Apparatus and methods for compensating for signal imbalance in a receiver

ABSTRACT

Apparatus, methods and systems for compensating for an I/Q imbalance may include compensating for an imbalance between a first component of a data signal and a second component of the data signal. The data signal may be modulated by a carrier signal having a frequency error. The first component may be characterized by at least one parameter. The method may include receiving the data and carrier signals; selecting a value for the parameter such that the frequency domain energy at negative frequencies is reduced; and modifying at least one of the components based on the value.

CROSS-REFERENCE TO OTHER APPLICATIONS

This is a nonprovisional of the following U.S. Provisional Applications, all of which are hereby incorporated by reference herein in their entireties: U.S. Provisional Application No. 60/866,532, entitled, “A METHOD FOR PACKET AGGREGATION IN A COORDINATED HOME NETWORK”, filed on Nov. 20, 2006, U.S. Provisional Application No. 60/866,527, entitled, “RETRANSMISSION IN COORDINATED HOME NETWORK” filed on Nov. 20, 2006, U.S. Provisional Application No. 60/866,519, entitled, “IQ IMBALANCE CORRECTION USING 2-TONE SIGNAL IN MULTI-CARRIER RECEIVERS”, filed on Nov. 20, 2006, U.S. Provisional Application No. 60/907,111, “SYSTEM AND METHOD FOR AGGREGATION OF PACKETS FOR TRANSMISSION THROUGH A COMMUNICATIONS NETWORK” filed on Mar. 21, 2007, U.S. Provisional Application No. 60/907,126, entitled, “MAC TO PHY INTERFACE APPARATUS AND METHODS FOR TRANSMISSION OF PACKETS THROUGH A COMMUNICATIONS NETWORK”, filed on Mar. 22, 2007, U.S. Provisional Application No. 60/907,819, entitled “SYSTEMS AND METHODS FOR RETRANSMITTING PACKETS OVER A NETWORK OF COMMUNICATION CHANNELS”, filed on Apr. 18, 2007, and U.S. Provisional Application No. 60/940,998, entitled “MOCA AGGREGATION”, filed on May 31, 2007.

FIELD OF THE INVENTION

The present invention relates generally to information networks and specifically to transmitting information such as media information over communication lines such as coax, thereby to form a communications network.

BACKGROUND OF THE INVENTION

Many structures, including homes, have networks based on coaxial cable (“coax”).

The Multimedia over Coax Alliance (“MoCA™”), provides at its website (www.mocalliance.org) an example of a specification (viz., that available under the trademark MoCA, which is hereby incorporated herein by reference in its entirety) for networking of digital video and entertainment information through coaxial cable. The specification has been distributed to an open membership.

Technologies available under the trademark MoCA, other specifications and related technologies (“the existing technologies”) often utilize unused bandwidth available on the coax. For example, coax has been installed in more than 70% of homes in the United States. Some homes have existing coax in one or more primary entertainment consumption locations such as family rooms, media rooms and master bedrooms. The existing technologies allow homeowners to utilize installed coax as a networking system and to deliver entertainment and information programming with high quality of service (“QoS”).

The existing technologies may provide high speed (270 mbps), high QoS, and the innate security of a shielded, wired connection combined with state of the art packet-level encryption. Coax is designed for carrying high bandwidth video. Today, it is regularly used to securely deliver millions of dollars of pay per view and premium video content on a daily basis. Networks based on the existing technologies can be used as a backbone for multiple wireless access points to extend the reach of wireless service in the structure.

Existing technologies provide throughput through the existing coaxial cables to the places where the video devices are located in a structure without affecting other service signals that may be present on the cable. The existing technologies provide a link for digital entertainment, and may act in concert with other wired and wireless networks to extend entertainment throughout the structure.

The existing technologies work with access technologies such as asymmetric digital subscriber lines (“ADSL”), very high speed digital subscriber lines (“VDSL”), and Fiber to the Home (“FTTH”), which provide signals that typically enter the structure on a twisted pair or on an optical fiber, operating in a frequency band from a few hundred kilohertz to 8.5 MHz for ADSL and 12 MHz for VDSL. As services reach such a structure via any type of digital subscriber line (“xDSL”) or FTTH, they may be routed via the existing technologies and the coax to the video devices. Cable functionalities, such as video, voice and Internet access, may be provided to the structure, via coax, by cable operators, and use coax running within the structure to reach individual cable service consuming devices in the structure. Typically, functionalities of the existing technologies run along with cable functionalities, but on different frequencies.

The coax infrastructure inside the structure typically includes coax, splitters and outlets. Splitters typically have one input and two or more outputs and are designed to transmit signals in the forward direction (input to output), in the backward direction (output to input), and to isolate outputs from different splitters, thus preventing signals from flowing from one coax outlet to another. Isolation is useful in order to a) reduce interference from other devices and b) maximize power transfer from Point Of Entry (“POE”) to outlets for best TV reception.

Elements of the existing technologies are specifically designed to propagate backward through splitters (“insertion”) and from output to output (“isolation”). One outlet in a structure can be reached from another by a single “isolation jump” and a number of “insertion jumps.” Typically isolation jumps have an attenuation of 5 to 40 dB and each insertion jump attenuates approximately 3 dB. MoCA™-identified technology has a dynamic range in excess of 55 dB while supporting 200 Mbps throughput. Therefore MoCA™-identified technology can work effectively through a significant number of splitters.

Managed network schemes, such as MoCA™-identified technology, are specifically designed to support streaming video with minimal packet loss between outlets.

When a network-connected device receives a data signal from the network, which may be a network such as that described above, the signal is often decomposed into in-phase (“I”) and quadrature (“Q”) portions during down-conversion to device base-band frequency. When the I and Q portions are recombined for data decryption, they are often imbalanced with respect to amplitude, phase or both. Rebalancing I and Q portions may involve calculating compensation factors based on frequency-domain signatures of the carrier frequency and the I and Q portions. In the presence of carrier frequency uncertainty, the frequency-domain signatures of received signals may be difficult to resolve using digital computation methods. It would therefore be desirable to provide systems and methods for compensating signals, in the presence of carrier frequency uncertainty, using digital computation methods.

SUMMARY OF THE INVENTION

A system and/or method for compensating for an I/Q imbalance at a node on a communication network, substantially as shown in and/or described in connection with at least one of the figures, as set forth more completely in the claims.

BRIEF DESCRIPTION OF THE DRAWINGS

The above and other features of the present invention, its nature and various advantages will be more apparent upon consideration of the following detailed description, taken in conjunction with the accompanying drawings, and in which:

FIG. 1 shows an illustrative schematic diagram of an illustrative single or multi-chip device that may be used in accordance with principles of the invention;

FIG. 2 shows an illustrative schematic diagram of a portion of a receiver in accordance with the principles of the invention;

FIG. 3 shows another illustrative schematic diagram of a portion of a receiver in accordance with the principles of the invention;

FIG. 4 shows an illustrative schematic diagram of a circuit in accordance with the principles of the invention;

FIG. 5 shows another illustrative schematic diagram of a circuit in accordance with the principles of the invention;

FIG. 6 shows an illustrative flow chart in accordance with the principles of the invention;

FIG. 7 shows, in abridged form, an illustrative data packet that may be processed in accordance with the principles of the invention;

FIG. 8 shows an illustrative portion of a discrete-valued frequency spectrum associated with signal processing in accordance with the principles of the invention;

FIG. 9 shows an illustrative schematic diagram of a another circuit in accordance with the principles of the invention;

FIG. 10 shows a schematic memory configuration in accordance with the principles of the invention:

FIG. 11 shows an illustrative Energy Loss of Image Figure as a function of the number of bins used;

FIG. 12A shows an illustrative simulation minimization and FIG. 12B shows various illustrative plots;

FIG. 13 shows an illustrative maximization plot;

FIG. 14 shows an illustrative plot that summarizes simulation results; and

FIG. 15 shows another illustrative plot that summarizes simulation results.

The worst case loss is experienced when the image falls midway between bins (r=1/(2N)). Using just one bin which is closest to the image results in a worst case loss of 3.9223 [dB] using two bins results in a loss of 0.9120 [dB] FIG. 11 summarizes the loss as a function of the number of bins used.

DETAILED DESCRIPTION OF EMBODIMENTS

Apparatus and methods for compensating for an I/Q imbalance are provided in accordance with the principles of the invention. The methods may include compensating for an imbalance between a first component of a data signal and a second component of the data signal. The data signal may be modulated by a carrier signal having a frequency error. The first component may be characterized by at least one parameter. The method may include receiving the data and carrier signals; selecting a value for the parameter such that a frequency domain energy is reduced, the frequency domain energy corresponding to a negative frequency; and modifying at least one of the components based on the selected value.

The apparatus may include a circuit operative to record signal values corresponding to frequency components of a received signal. The signal may be one that carries at least one orthogonal frequency division multiplexing (“OFDM”) symbol. The signal values may correspond to a carrier frequency having a frequency error; a first tone; and a second tone.

The apparatus may include a system for compensating for an imbalance between a first component of a data signal and a second component of the data signal. The data signal may be modulated by a carrier signal having a frequency error. The first component may be characterized by at least one parameter. The system may include a hardware module configured to quantify a signal value corresponding to one of the data and carrier signals; and a software module configured to receive the signal value from the hardware.

The first and second tones may be transmitted in the context of a MoCA protocol probe2 transmission as set forth in the aforementioned MoCA specification.

Illustrative features of the invention are described below with reference to FIGS. 1-8 and Appendices A-E.

FIG. 1 shows a single or multi-chip module 102 according to the invention, which can be one or more integrated circuits, in an illustrative data processing system 100 according to the invention. Data processing system 100 may include one or more of the following components: I/O circuitry 104, peripheral devices 106, processor 108 and memory 110. These components may be coupled together by a system bus or other interconnections 112 and are disposed on a circuit board 120 in an end-user system 130. Elements of module 102 may perform tasks involved in I/Q imbalance compensation.

In some embodiments, I/Q imbalance compensation may be performed during MoCA Probe2 burst reception. Probe2 is a two-tone signal which can be used for I/Q imbalance calculations or other RF calibrations in the receiver. A PHY layer performs bin selection and recording and the result is uploaded to the CPU for the I/Q compensation parameters calculations.

FIG. 2 shows a partial schematic diagram of illustrative receiver 200. Receiver 200 may include radio frequency (“RF”) processing module 202, time domain processing module 204 and frequency domain processing module 206. RF signal 208 is received and gain-adjusted at gain 210. Signal 208 is down-converted to base band (“BB”) frequency at 212. Intentional frequency error 213 is added to signal 208 at 212. Analog-to-digital converter 214 converts signal 208 to a digital signal sampled at the analog-to-digital sampling rate and passes signal 208 to imbalance compensation module 218. I/Q imbalance compensation module 218 may be configured to carry out steps associated herein with I/Q compensation. I/Q imbalance compensation module 218 outputs signal 209, which corresponds to Equation 1 (below).

Signal 209 passes to variable rate interpolator 224, which resamples signal 209 to an appropriate sampling rate.

The variable rate interpolator 224 may receive timing signal 237 from numerically controlled oscillator (“NCO”) timing generator 236. Timing signal 237 may be based on carrier frequency offset estimate (“CFOE”) 241, from preamble processor 240. CFOE 241 may be based on a preamble processor 240 estimate. Interpolator 224 outputs signal 225, which may then pass through high pass filter (“HPF”) 228 to reject direct current (“DC”) signal components.

Carrier recovery loop 229 may be present to perform frequency compensation for intentional frequency error 213. Carrier recovery loop 229 may receive input from NCO frequency generator 234, which may be controlled by receiver controller 232. NCO frequency generator 234 may receive carrier frequency offset estimate 241 from preamble processor 240. A cyclic prefix may be removed from signal 225 at CP remover 246.

Fast Fourier transform module 298 may be present in frequency domain processing module 206 to transform signal 225 into frequency domain information (“FFT output”) that may be stored in memory 299 and may be communicated to probe2 software processing routine 250, which may output correction parameters 252 for return to I/Q imbalance compensation module 218.

FIG. 3 shows a partial schematic diagram of illustrative receiver 300. Receiver 300 may include radio frequency (“RF”) processing module 302, time domain processing module 304 and frequency domain processing module 306. RF signal 308 is received and gain-adjusted at gain 310. Signal 308 is down-converted to base band frequency at 312. Intentional frequency error 313 is added to signal 308 at 312. Analog-to-digital converter 314 converts signal 308 to a digital signal and passes signal 308 to 100 MHz FIFO (“first in, first out”) buffer 316. Buffer 316 passes signal 308 to I/Q imbalance compensation module 318. I/Q imbalance compensation module 318 may be configured to carry out steps associated herein with I/Q compensation. I/Q imbalance compensation module 318 outputs signal 309, which corresponds to Equation 1 (below).

Signal 309 passes to baseband-mode demixer 320. Receiver 300 may include automatic gain controller 322, which may provide feedback to gain 310 based on signal 309. From demixer 320, signal 309 may pass to Farrow interpolator 324, which resamples 100 MHz signal 309 at a lower rate.

Farrow interpolator 324 may receive timing signal 337 from numerically controlled oscillator (“NCO”) timing generator 336. Timing signal 337 may be based on carrier frequency offset estimate 341, from preamble control processor 340. Carrier frequency offset estimate 341 may be based on the output of TD phase rotator 330 (discussed below), via preamble processor 340. In some embodiments, interpolator 324 outputs signal 325 at 100 MHz. Signal 325 may be synchronized to a transmitter clock (not shown) via a timing recover loop (not shown). Signal 325 may be down-sampled by a factor of 2, via half band filter decimator (“HB DEC 2→1”) 326, to 50 MHz. Signal 325 may then pass through high pass filter (“HPF”) 328 to reject direct current (“DC”) signal components.

Time domain (“TD”) phase rotator 330 may be present to perform frequency compensation for intentional frequency error 313. TD phase rotator may receive input from NCO frequency generator 334, which may be controlled by receiver controller 332. NCO frequency generator 334 may receive carrier frequency offset estimate 341 from preamble processor 340. Signal 325 may then pass to delay buffer 342. A cyclic prefix may be removed at sub-circuit 346. In some embodiments, sub-circuit 346 may perform receiver windowing to reduce damage from narrow band interference noise that might otherwise leak into adjacent tones.

Fast Fourier transform module 398 may be present in frequency domain processing module 306 to transform signal 325 into frequency domain information that may be communicated to probe2 calculator 350, which may output probe2 result 352, for transmission to I/Q compensation module 318.

Some embodiments include a bypass mode, in which signal input is routed to output around I/Q imbalance compensation module 318.

In some embodiments, I/Q compensation is accomplished by digital signal analysis and processing. In those embodiments, ζ, ρ & Scale_Q are I/Q compensation parameters that have to be estimated during Probe2.

Equation 1 shows compensated real and imaginary portions of a compensated signal that would be output from the I/Q imbalance compensation. module (see FIG. 2).

$\begin{matrix} {\mspace{79mu}{{\overset{\sim}{Y}}_{real} = \left\{ {{\begin{matrix} {{Bypass}==1} & Y_{real} \\ {{Bypass}==0} & \left\{ \begin{matrix} {{Scale\_ Q}==0} & {\varsigma\; Y_{real}} \\ {{Scale\_ Q}==1} & Y_{real} \end{matrix} \right. \end{matrix}{\overset{\sim}{Y}}_{imag}} = \left\{ \begin{matrix} {{Bypass}==1} & Y_{imag} \\ {{Bypass}==0} & \left\{ \begin{matrix} {{Scale\_ Q} = 0} & {{\overset{\sim}{Y}}_{imag} = {Y_{imag} + {\rho\; Y_{real}}}} \\ {{Scale\_ Q} = 1} & {{\varsigma\; Y_{imag}} + {\rho\; Y_{real}}} \end{matrix} \right. \end{matrix} \right.} \right.}} & {{Equation}\mspace{20mu} 1} \end{matrix}$

FIG. 4 shows illustrative circuit 400, which may be included in a device for implementing the compensation set forth in Equation 1.

FIG. 5 shows illustrative circuit 500, which may be included in a device for implementing the compensation set forth in Equation 1. Appendix A sets forth the theoretical basis for the compensation set forth in Equation 1. Appendix B shows exemplary imbalance and compensation measurements that have been made in connection with the apparatus and methods described herein.

FIG. 6 shows illustrative process 600 for compensating I/Q imbalance. Process 600 involves both hardware (“HW”) and software (“SW”) operations. The output of initial hardware operation 602 is a data array Z(k,m), which is the output of a Fast Fourier Transform (“FFT”) at bin k corresponding to Probe2 OFDM symbol m.

Initial hardware operation 602 may include numerically controlled oscillator (“NCO”) phase reset 604. The phase of the first sample of an FFT window that results from Time Domain Unit (“TDU”) frequency compensation is determined. For this purpose the NCO Phase of the Phase Rotator in the Receiver TDU shall be reset to zero anytime after fine frequency compensation has been computed. The number of samples (number of phase accumulations) between the reset of the NCO and the first sample of the FFT window denoted as Δn shall be computed and sent to the SW routine. Zero phase accumulations (i.e., Δn=0) is most desirable since it reduces complexity of the SW routine. For the setting Δn=0, NCO phase accumulator 335 (in NCO frequency generator 334—see FIG. 3) should be reset once the first sample of the 356 point FFT window propagates through TD phase rotator 330 (see FIG. 3) (and thus the first sample would be multiplied by unity).

FIG. 7 shows packet 700, NCO reset, Δn and the start of the FFT window.

In some embodiments, bin selection 606 (see FIG. 6) may be performed as a floating point computation, in which i₁ and i₂ are frequency bin indices computed as shown in Equation 2.

$\begin{matrix} {{i_{2} = {- {{round}\left( \frac{N \cdot {CFO}}{\pi} \right)}}}{i_{2} = {i_{1} + {{{sign}({CFO})} \cdot {{sign}\left( {{i_{1}} - {\frac{N \cdot {CFO}}{\pi}}} \right)}}}}} & {{Equation}\mspace{20mu} 2} \end{matrix}$ Wherein CFO/(2π) is the estimated carrier frequency offset between a transmitter and the receiver and N is the number of FFT bins (e.g., 256).

In some embodiments, bin selection 606 (see FIG. 6) may be performed as a fixed point computation. In those embodiments, CFO is a 17 bit signed integer, where ‘1’=2¹⁴. The computation of i₁ and i₂ may be done via comparison to fixed thresholds. The value of the FFT grid in fixed point representation is given by Equation 3:

$\begin{matrix} {{F_{k} = {{round}\left( {\frac{2\pi\; k}{N} \cdot 2^{{{Freq}\_{bits}} - 1}} \right)}},{{{for}\mspace{14mu} k} = \left\lbrack {{- 3},3} \right\rbrack},} & {{Equation}\mspace{20mu} 3} \end{matrix}$ in which Freq_bits may be set to 14 or any other suitable number. Indices i₁ and i₂ are selected by finding the two FFT bins closest to 2CFO.

FIG. 8 shows decision areas (absolute values only due to symmetry) corresponding to Equation 3.

FIG. 9 shows an illustrative hardware (“HW”) implementation for decision area boundary selection. Table 1 shows illustrative boundary values.

TABLE 1 |Frequency Boundaries| $\frac{F_{0} + F_{1}}{2}$ $\frac{F_{1} + F_{2}}{2}$ $\frac{F_{2} + F_{3}}{2}$ F₁ F₂ F₃ Fixed point value 804 2413 4012 1608 3217 4825

Equation 4 sets forth a definition for the sign operation.

$\begin{matrix} {{{sign}(x)} = \left\{ \begin{matrix} 1 & {x \geq 0} \\ {- 1} & {x < 0} \end{matrix} \right.} & {{Equation}\mspace{20mu} 4} \end{matrix}$

In some embodiments, bin recording (at step 408, see FIG. 4) may involve 16 bit FFT outputs at bins k₁, −k₁+i₁, −k₁+i₂, k₂, −k₂+i₁, −k₂+i₂, which are then recorded for each of L OFDM symbols. It will be understood that there may be any suitable number of bits at the FFT output. The addresses in MoCA FFT that correspond to the bins are set forth in Table 2.

TABLE 2 ADDR: k₁ Z[k₁, m] k₁ ∈ [146, 186] ADDR: k₂ Z[k₂, m] k₂ ∈ [217, 249] ADDR: 256 − Z[−k₁ + i₁, m], −k₁ + i₁ = 256 − k₁ + i₁ ∈ [67, 113] k₁ + i₁ ADDR: 256 − Z[−k₁ + i₂, m], −k₁ + i₂ = 256 − k₁ + i₂ ∈ [66, 112] k₁ + i₂ ADDR: 256 − Z[−k₂ + i₁, m], −k₂ + i₁ = 256 − k₂ + i₁ ∈ [4, 36] k₂ + i₁ ADDR: 256 − Z[−k₂ + i₂, m] −k₂ + i₂ = 256 − k₂ + i₂ ∈ [3, 37] k₂ + i₂

FIG. 10 shows an illustrative memory map at the end of a probe2 burst.

A CFO estimate is recorded at step 608 (shown in FIG. 6). The estimate may be a 17-bit estimate.

In some embodiments, residual frequency error {circumflex over (ε)} estimation 610 (see FIG. 6) may be performed by a software module. In some embodiments, residual frequency error estimation may be performed by a hardware module. In some embodiments, residual frequency error estimation may be computed as shown in Equation 5.

$\begin{matrix} {{\hat{ɛ} = \frac{{angle}\left( {\sum\limits_{m = 0}^{L - 2}{{Z\left\lbrack {k_{i},m} \right\rbrack}{Z^{*}\left\lbrack {k_{i},{m + 1}} \right\rbrack}}} \right)}{2{\pi\left( {N + N_{CP}} \right)}}}{Where}{k_{i} = \left\{ \begin{matrix} k_{1} & {{SNR}_{k_{1}} > {SNR}_{k_{2}}} \\ k_{2} & {{SNR}_{k_{2}} > {SNR}_{k_{1}}} \end{matrix} \right.}} & {{Equation}\mspace{20mu} 5} \end{matrix}$

In some embodiments, residual frequency error compensation and time averaging may be computed in accordance with Equations 6, which depend on {circumflex over (ε)} and whose derivations are set forth in Appendix A.

$\begin{matrix} {{{\overset{\_}{Z}}_{k_{1}} = {\sum\limits_{m = 0}^{L - 1}{{Z\left\lbrack {k_{1},m} \right\rbrack}{\mathbb{e}}^{{- {j2\pi}}\;{\hat{ɛ}{({N + N_{CP}})}}m}}}}{{\overset{\_}{Z}}_{k_{2}} = {\sum\limits_{m = 0}^{L - 1}{{Z\left\lbrack {k_{2},m} \right\rbrack}{\mathbb{e}}^{{- {j2\pi}}{\hat{ɛ}{({N + N_{CP}})}}m}}}}{{\overset{\_}{Z}}_{{- k_{1}} + i_{1}} = {\sum\limits_{m = 0}^{L - 1}{{Z\left\lbrack {{{- k_{1}} + i_{1}},m} \right\rbrack}{\mathbb{e}}^{{- {j{({{2{CFO}} + {2\pi\;\hat{ɛ}}})}}}{({N + N_{CP}})}m}}}}{{\overset{\_}{Z}}_{{- k_{2}} + i_{1}} = {\sum\limits_{m = 0}^{L - 1}{{Z\left\lbrack {{{- k_{2}} + i_{1}},m} \right\rbrack}{\mathbb{e}}^{{- {j{({{2{CFO}} + {2\pi\;\hat{ɛ}}})}}}{({N + N_{CP}})}m}}}}{{\overset{\_}{Z}}_{{- k_{1}} + i_{2}} = {\sum\limits_{m = 0}^{L - 1}{{Z\left\lbrack {{{- k_{1}} + i_{2}},m} \right\rbrack}{\mathbb{e}}^{{- {j{({{2{CFO}} + {2\pi\;\hat{ɛ}}})}}}{({N + N_{CP}})}m}}}}{{\overset{\_}{Z}}_{{- k_{2}} + i_{2}} = {\sum\limits_{m = 0}^{L - 1}{{Z\left\lbrack {{{- k_{2}} + i_{2}},m} \right\rbrack}{\mathbb{e}}^{{- {j{({{2{CFO}} + {2\pi\;\hat{ɛ}}})}}}{({N + N_{CP}})}m}}}}} & {{Equations}\mspace{20mu} 6} \end{matrix}$

Equations 7 may be used to evaluate an I/Q imbalance phasor estimate, which may be computed using Equation 8.

$\begin{matrix} {B_{n} = \left\{ {{\begin{matrix} {{CFO} \neq 0} & {{\frac{1}{2}\left( \frac{\sin\left( {2 \cdot {CFO} \cdot N} \right)}{{CFO} + \frac{\pi \cdot i_{n}}{N}} \right)} +} \\ \; & {\frac{j}{2}\left( \frac{{\cos\left( {2 \cdot {CFO} \cdot N} \right)} - 1}{{CFO} + \frac{\pi \cdot i_{n}}{N}} \right)} \\ {{{CFO} = 0},{i_{n} = 0}} & N \\ {{{CFO} = 0},{i_{n} \neq 0}} & 0 \end{matrix}C_{1}} = {{\frac{{\mathbb{e}}^{j\; 2{{CFO}{({\Delta\; n})}}}}{N}\frac{\left( {{B_{1}}^{2} + {B_{2}}^{2}} \right) \cdot {\overset{\_}{Z}}_{k_{1}}}{{B_{1}\left( {\overset{\_}{Z}}_{{- k_{1}} + i_{1}} \right)}^{*} + {B_{2}\left( {\overset{\_}{Z}}_{{- k_{1}} + i_{2}} \right)}^{*}}C_{2}} = {{\frac{{\mathbb{e}}^{j\; 2{{CFO}{({\Delta\; n})}}}}{N}\frac{\left( {{B_{1}}^{2} + {B_{2}}^{2}} \right) \cdot {\overset{\_}{Z}}_{k_{2}}}{{B_{1}\left( {\overset{\_}{Z}}_{{- k_{2}} + i_{1}} \right)}^{*} + {B_{2}\left( {\overset{\_}{Z}}_{{- k_{2}} + i_{2}} \right)}^{*}}C} = {\frac{1}{2}\left( {C_{1} + C_{2}} \right)}}}} \right.} & {{Equations}\mspace{20mu} 7} \\ {\overset{\_}{g\;{\mathbb{e}}^{{- j}\; 0}} = \frac{C - 1}{C - 1}} & {{Equation}\mspace{20mu} 8} \end{matrix}$

I/Q imbalance compensation parameters ξ, ρ and Scale_Q (see, e.g., Equation 1) may then be computed in accordance with Equation 9.

$\begin{matrix} \left\{ \begin{matrix} {{{real}\left\{ \overset{\_}{g\;{\mathbb{e}}^{- {j\theta}}} \right\}} \geq 1} & {{ScaleQ} = 1} & {{\hat{\xi} = \frac{1}{{real}\left\{ \overset{\_}{g\;{\mathbb{e}}^{- {j\theta}}} \right\}}},} & {\hat{\rho} = {- \frac{{imag}\left\{ \overset{\_}{g\;{\mathbb{e}}^{- {j\theta}}} \right\}}{{real}\left\{ \overset{\_}{g\;{\mathbb{e}}^{- {j\theta}}} \right\}}}} \\ {otherwise} & {{ScaleQ} = 0} & {{\hat{\xi} = \left\{ \overset{\_}{g\;{\mathbb{e}}^{- {j\theta}}} \right\}},} & {\hat{\rho} = {{- {imag}}\left\{ \overset{\_}{g\;{\mathbb{e}}^{- {j\theta}}} \right\}}} \end{matrix} \right. & {{Equation}\mspace{20mu} 9} \end{matrix}$

Equation 9 avoids saturation at the receiver since ξ is always smaller or equal than unity, thus attenuating the stronger I/Q signal rather than amplifying the weaker I/Q signal. In some embodiments, the above computations can be carried out in an iterative fashion over several probe2 transmissions. Equations 10 show how new phasor estimates may be used to update previous estimates.

$\begin{matrix} \begin{matrix} {\left( \overset{\_}{g\;{\mathbb{e}}^{- {j\theta}}} \right)_{i}^{ACC} = {{\left( {1 - \mu} \right)\left( \overset{\_}{g\;{\mathbb{e}}^{- {j\theta}}} \right)_{i - 1}^{ACC}} +}} \\ {{\mu_{i}\left( \overset{\_}{g\;{\mathbb{e}}^{- {j\theta}}} \right)}_{i - 1}^{ACC} \cdot \left( \overset{\_}{g\;{\mathbb{e}}^{- {j\theta}}} \right)_{i}} \\ {\left( \overset{\_}{g\;{\mathbb{e}}^{- {j\theta}}} \right)_{0}^{ACC} = 1} \end{matrix} & {{Equations}\mspace{20mu} 10} \end{matrix}$

In Equations 10,

$\left( \overset{\_}{g\;{\mathbb{e}}^{- {j\theta}}} \right)_{i}.$ is the phasor estimate computed during the i'th probe2 transmission. Some embodiments may include an update routine that may use a first order loop with a loop gain of μ_(i)ε[0,1]. The loop gain may provide a tradeoff between convergence speed and noise filtering by controlling the loop bandwidth (“BW”). A gear-shifting approach may be used in which the loop BW is dynamically changed during convergence. For fast convergence during the first two/three iterations, a high loop BW may be used. For consecutive probe2 transmissions, a small loop BW may be used. Equation 11 sets forth values that may be used for μ_(i). i denotes the probe2 burst index number.

$\begin{matrix} {\mu_{i} = \left\{ \begin{matrix} 1 & {i = 1} \\ 0.75 & {i = 2} \\ 0.5 & {i = 3} \\ 0.25 & {i = 4} \end{matrix}\quad \right.} & {{Equation}\mspace{20mu} 11} \end{matrix}$

Equations 12 set forth I/Q compensation parameters that may be used during the reception of the i'th probe2.

$\begin{matrix} \left\{ \begin{matrix} {{{real}\left\{ \left( \overset{\_}{g\;{\mathbb{e}}^{- {j\theta}}} \right)_{i - 1}^{ACC} \right\}} \geq 1} & {\hat{\xi} = \frac{1}{{real}\left\{ \left( \overset{\_}{g\;{\mathbb{e}}^{- {j\theta}}} \right)_{i - 1}^{ACC} \right\}}} & {{\hat{\rho}}_{i} = {- \frac{{imag}\left\{ \left( \overset{\_}{g\;{\mathbb{e}}^{- {j\theta}}} \right)_{i - 1}^{ACC} \right\}}{{real}\left\{ \left( \overset{\_}{g\;{\mathbb{e}}^{- {j\theta}}} \right)_{i - 1}^{ACC} \right\}}}} \\ {otherwise} & {{\hat{\xi} = {{real}\left\{ \left( \overset{\_}{g\;{\mathbb{e}}^{- {j\theta}}} \right)_{i - 1}^{ACC} \right\}}},} & {\hat{\rho_{i}} = {{- {imag}}\left\{ \left( \overset{\_}{g\;{\mathbb{e}}^{- {j\theta}}} \right)_{i - 1}^{ACC} \right\}}} \end{matrix} \right. & {{Equation}\; s\mspace{14mu} 12} \end{matrix}$

Three to four iterations (which may correspond to 3 to 4 probe2 transmissions) are often sufficient to compensate for I/Q imbalance.

Appendix C sets forth pseudo-code for a fixed point implementation of the compensation.

Appendix D sets forth parameters for a hardware-software interface in a system for I/Q imbalance compensation.

A network node may acquire an estimate of signal to noise ratio (“SNR”) at each tone and carrier frequency offset (relative to an associated network coordinator (“NC”)) when the node processes one or more probe 1 bursts from the NC. The node may use the SNR estimates to inform the NC which two frequency bins to use for probe2 transmission to the node. The node may use the CFO estimate to calculate and communicate to the NC the number of OFDM symbols and the cyclic prefix (“CP”) length during probe2 transmission.

Appendix E sets forth illustrative pseudocode for computation of frequency offset introduction, CP and selection of a number of OFDM symbols. In some embodiments, the Probe2, CP and L algorithms set forth in Appendix E may be performed before sending a MoCA™ probe2 report and after a receiver RF generator introduces any required, necessary or intentional carrier offset.

For the sake of clarity, the foregoing description, including specific examples of parameters or parameter values, is sometimes specific to certain protocols such as those identified with the name MoCA™ and/or Ethernet protocols. However, this is not intended to be limiting and the invention may be suitably generalized to other protocols and/or other packet protocols. The use of terms that may be specific to a particular protocol such as that identified by the name MoCA™ or Ethernet to describe a particular feature or embodiment is not intended to limit the scope of that feature or embodiment to that protocol specifically; instead the terms are used generally and are each intended to include parallel and similar terms defined under other protocols.

It will be appreciated that software components of the present invention including programs and data may, if desired, be implemented in ROM (read only memory) form, including CD-ROMs, EPROMs and EEPROMs, or may be stored in any other suitable computer-readable medium such as but not limited to discs of various kinds, cards of various kinds and RAMs. Components described herein as software may, alternatively, be implemented wholly or partly in hardware, if desired, using conventional techniques.

Thus, systems and methods for compensating for I/Q imbalance have been described. Persons skilled in the art will appreciate that the present invention can be practiced using embodiments of the invention other than those described, which are presented for purposes of illustration rather than of limitation. The present invention is limited only by the claims which follow.

APPENDIX A

Probe2 Theory

The I/Q imbalance can be modeled as a multiplicative gain factor applied on one of the I/Q components as well a relative phase difference. During probe2 reception MoCA specifies that a receiver must introduce a frequency error during RF down conversion we shall denote this shift as φ. The converted signal is given by: z _(i) [n]=s _(i) [n] cos(2πφn)−s _(q) [n] sin(2πφn)+w _(i) [n] z _(q) [n]=gs _(i) [n] sin(2πφn−θ)+gs _(q) [n] cos(2πφn−θ)+w _(q) [n]

Some algebra shows that the above can be expressed as

z[n] = K₁s[n]𝕖^(j2πφ n) + K₂s * [n]𝕖^(−j2πφ n) + w[n] $K_{1} = {\frac{1}{2}\left( {1 + {g\;{\mathbb{e}}^{- {j\theta}}}} \right)}$ $K_{2} = {\frac{1}{2}\left( {1 - {g\;{\mathbb{e}}^{- {j\theta}}}} \right)}$

At the receiver I/Q compensation is performed the signal after I/Q compensation is given by:

z^(′)[n] = {real(K₁s[n]𝕖^(j2πφ n) + K₂s * [n]𝕖^(−j2πφ n) + w[n])} + jξ{imag(K₁s[n]𝕖^(j2πφ n) + K₂s * [n]𝕖^(−j2πφ n) + w[n])} + j ρ{real(K₁s[n]𝕖^(j2πφ n) + K₂s * [n]𝕖^(−j2πφ n) + w[n])}

Assuming transmission of a single frequency at frequency bin k, after some algebra the compensated signal is given by

${z^{\prime}\lbrack n\rbrack} = {\left\{ {{h}{\cos\left( {{2\pi\;{n\left( {\frac{k}{N} + \varphi} \right)}} + {\angle\; h}} \right)}} \right\} + {{j\xi}\left\{ {g{h}\left( {{2\pi\;{n\left( {\frac{k}{N} + \varphi} \right)}} + {\angle\; h} - \theta} \right)} \right\}} + {{j\rho}\left\{ {{h}{\cos\left( {{2\pi\;{n\left( {\frac{k}{N} + \varphi} \right)}} + {\angle\; h}} \right)}} \right\}} + \left\lbrack {{w_{r}\lbrack n\rbrack} + {j\left( {{\rho \cdot {w_{r}\lbrack n\rbrack}} + {\xi \cdot {w_{i}\lbrack n\rbrack}}} \right)}} \right\rbrack}$

The signal over goes frequency compensation and then is transformed into the frequency domain via the FFT operation. After some algebra the frequency domain signals at bins k and −k are given by:

$\begin{matrix} {{\hat{Z}\lbrack k\rbrack} = {{{h}{\mathbb{e}}^{{j\angle}\; h}{N\left\lbrack {{\frac{1}{2}\left( {1 + {\xi\; g\;{\mathbb{e}}^{- {j\theta}}}} \right)} + {\frac{\rho}{2}{\mathbb{e}}^{j\;\frac{\pi}{2}}}} \right\rbrack}} +}} \\ {{h}{\mathbb{e}}^{{- {j\angle}}\; h}{\mathbb{e}}^{- {j{({2{\pi{({N - 1})}}{({\frac{k}{N} + \varphi})}})}}}{\frac{\sin\left( {2{\pi\left( {k + {N\;\varphi}} \right)}} \right)}{\sin\left( {2{\pi\left( {\frac{k}{N} + \varphi} \right)}} \right)}\left\lbrack {{\frac{1}{2}\left( {1 - {\xi\; g\;{\mathbb{e}}^{j\theta}}} \right)} + {\frac{\rho}{2}{\mathbb{e}}^{j\;\frac{\pi}{2}}}} \right\rbrack}} \end{matrix}$ $\mspace{45mu}{{\hat{Z}\left\lbrack {- k} \right\rbrack} = {{h}{\mathbb{e}}^{{- {j\angle}}\; h}{\mathbb{e}}^{- {j{({2{{\pi\varphi}{({N - 1})}}})}}}\frac{\sin\left( {2\pi\;\varphi\; N} \right)}{\sin\left( {2{\pi\varphi}} \right)}\left\{ {{\frac{1}{2}\left( {1 - {\xi\; g\;{\mathbb{e}}^{j\theta}}} \right)} + {\frac{\rho}{2}{\mathbb{e}}^{j\;\frac{\pi}{2}}}} \right\}}}$

In a system without I/Q imbalance, the energy at the negative bin is zero. The energy at the negative bin due to I/Q imbalance is given by

${{\hat{Z}\left\lbrack {- k} \right\rbrack}}^{2} = {{{h}^{2}\left( \frac{\sin\left( {2\pi\;\varphi\; N} \right)}{\sin\left( {2{\pi\varphi}} \right)} \right)^{2}\left\{ {{\frac{1}{2}\left( {1 - {\xi\; g\;{\mathbb{e}}^{j\theta}}} \right)} + {\frac{\rho}{2}{\mathbb{e}}^{j\;\frac{\pi}{2}}}} \right\}\left\{ {{\frac{1}{2}\left( {1 - {\xi\; g\;{\mathbb{e}}^{- {j\theta}}}} \right)} + {\frac{\rho}{2}{\mathbb{e}}^{{- j}\;\frac{\pi}{2}}}} \right\}} = {\frac{1}{4}{h}^{2}\left( \frac{\sin\left( {2\pi\;\varphi\; N} \right)}{\sin\left( {2{\pi\varphi}} \right)} \right)^{2}\left\{ {1 + {\xi^{2}g^{2}} + \rho^{2} - {2\xi\; g\;\cos\;\theta} - {2\xi\; g\;\rho\;{\sin(\theta)}}} \right\}}}$

Thus our target is to minimize the energy of bin −k by using ρ, ξ. Minimizing using the Lagrange multipliers method gives the following equations

$\frac{\partial{{\hat{Z}\left\lbrack {- k} \right\rbrack}}^{2}}{\partial\xi} = {\frac{\partial\left( {1 + {\xi^{2}g^{2}} + \rho^{2} - {2\xi\; g\;\cos\;\theta} - {2\xi\; g\;\rho\;{\sin(\theta)}}} \right)}{\partial\xi} = 0}$ $\frac{\partial{{\hat{Z}\left\lbrack {- k} \right\rbrack}}^{2}}{\partial\rho} = {\frac{\partial\left( {1 + {\xi^{2}g^{2}} + \rho^{2} - {2\xi\; g\;\cos\;\theta} - {2\xi\; g\;\rho\;{\sin(\theta)}}} \right)}{\partial\rho} = 0}$ 2ξ g² − 2g cos  θ − 2ρ g sin (θ) = 0 2ρ − 2ξ g sin (θ) = 0

Solving the above yields

${\begin{bmatrix} g^{2} & {{- g}\;\sin\;(\theta)} \\ {{- g}\;{\sin(\theta)}} & 1 \end{bmatrix}\begin{bmatrix} \xi \\ \rho \end{bmatrix}} = {{\begin{bmatrix} {g\;\cos\;\theta} \\ 0 \end{bmatrix}\begin{bmatrix} \xi \\ \rho \end{bmatrix}} = {{\frac{1}{\left( {g\;\cos\;\theta} \right)^{2}}\begin{bmatrix} {g\;\cos\;\theta} \\ {g^{2}\sin\;\theta\;\cos\;\theta} \end{bmatrix}} = \left\lbrack \frac{1}{\begin{matrix} {g\;\cos\;\theta} \\ {\tan\;\theta} \end{matrix}} \right\rbrack}}$

It is easy to show that such a selection actually brings the energy at bin −k to 0 and thus completely cancels the I/Q imbalance effects. Our goal now is to estimate the I/Q imbalance parameters from probe2 transmissions.

I/Q Parameter Estimation

Since I/Q imbalance corrupts the incoming signal it results in corrupted carrier frequency estimation as well as corrupted channel estimation. The channel estimation under I/Q imbalance is given by;

${\hat{h}}_{k} = {{\frac{A_{k}^{*}}{N{A_{k}}^{2}}{\hat{Z}\lbrack k\rbrack}} = {{{h}{\mathbb{e}}^{{j\angle}\; h}K_{1}} + e_{k}}}$

The FFT output at bins k and −k without I/Q compensation but after frequency compensation assuming a frequency estimation error of ε is given by

${Z\lbrack k\rbrack} = {{{h}{\mathbb{e}}^{\angle\; h}K_{1}{\mathbb{e}}^{{j\pi ɛ}{({N - 1})}}\frac{\sin\left( {{\pi ɛ}\; N} \right)}{\sin({\pi ɛ})}} + {{h}{\mathbb{e}}^{{- \angle}\; h}K_{2}{\mathbb{e}}^{{- {{j2\pi}{({N - 1})}}}{({\frac{k}{N} + \varphi - {\frac{1}{2}ɛ}})}}\frac{\sin\left( {2\pi\;{N\left( {\frac{k}{N} + \varphi - {\frac{1}{2}ɛ}} \right)}} \right)}{\sin\left( {2\pi\;\left( {\frac{k}{N} + \varphi - {\frac{1}{2}ɛ}} \right)} \right)}} + W_{k}}$ ${Z\left\lbrack {- k} \right\rbrack} = {{{h}{\mathbb{e}}^{{- \angle}\; h}K_{2}{{\mathbb{e}}^{{- {{j2\pi}{({\varphi - {\frac{1}{2}ɛ}})}}}{({N - 1})}}\left( \frac{\sin\left( {2\pi\;{N\left\lbrack {\varphi - {\frac{1}{2}ɛ}} \right\rbrack}} \right)}{\sin\left( {2{\pi\left\lbrack {\varphi - {\frac{1}{2}ɛ}} \right\rbrack}} \right)} \right)}} + {{h}{\mathbb{e}}^{\angle\; h}K_{1}{{\mathbb{e}}^{{- {{j2\pi}{({\frac{k}{N} + {\frac{1}{2}ɛ}})}}}{({N - 1})}}\left( \frac{\sin\left( {2\pi\;{N\left\lbrack {\frac{k}{N} + {\frac{1}{2}ɛ}} \right\rbrack}} \right)}{\sin\left( {2{\pi\left\lbrack {\frac{k}{N} + {\frac{1}{2}ɛ}} \right\rbrack}} \right)} \right)}} + W_{- k}}$ Effects of Carrier Frequency Offset Greater than 50 Khz

In the absence of carrier frequency error the image component resulting from the I/Q imbalance appears exactly at the mirror digital frequency (−k/N) of the transmitted tone. Under carrier frequency error (which is mandatory during probe II) the I/Q image appears at a digital frequency of (−k/N−2φ), where φ is the normalized carrier frequency error which is φ=Carrier Frequency Error/SymbolRate=Δfc/fs. The carrier frequency error can be as large as ±200 ppm of 1.5e9 Hz=300 kkHz. While the OFDM tone spacing is 50e6/256=195.3 kHz. Thus the image component can fall somewhere between [−k−3, k+3] interval in the frequency domain. The FFT output for bin −k+i is given by

${Z\left\lbrack {{- k} + i} \right\rbrack} = {{{h}{\mathbb{e}}^{{- \angle}\; h}K_{2}{\mathbb{e}}^{{- {{j2\pi}{({\varphi - {\frac{1}{2}ɛ} + \frac{i}{2N}})}}}{({N - 1})}}\left( \frac{\left( {- 1} \right)^{i}{\sin\left( {2\pi\;{N\left\lbrack {\varphi - {\frac{1}{2}ɛ}} \right\rbrack}} \right)}}{\sin\left( {2{\pi\left\lbrack {\varphi - {\frac{1}{2}ɛ} + \frac{i}{2N}} \right\rbrack}} \right)} \right)} + {{h}{\mathbb{e}}^{\angle\; h}K_{1}{{\mathbb{e}}^{{- {{j\pi}{({\frac{{2k} - i}{N} + ɛ})}}}{({N - 1})}}\left( \frac{\sin\left( {\pi\;{N\left\lbrack {\frac{{2k} - i}{N} + ɛ} \right\rbrack}} \right)}{\sin\left( {\pi\;\left\lbrack {\frac{{2k} - i}{N} + ɛ} \right\rbrack} \right)} \right)}} + W_{{- k} + i}}$

And so due to the fact that the compensated frequency error φ results in a shift of 2φ in the location of the image, we need to collect the image energy from the interval [−k−3, . . . , k+3]. Pragmatically since we know the frequency error φ (up to ε) we know that the image will appear at a digital frequency of

${- \left( {\frac{k + M}{N} + r} \right)},{{2\varphi} = {{\frac{M}{N} + {r\mspace{14mu}{where}\mspace{14mu} M}} = {{\left\lfloor {2\varphi\; V} \right\rfloor\mspace{14mu}{and}\mspace{14mu}{r}} < {\frac{1}{N}.}}}}$

The loss of image energy in [dB] with respect to the image energy is a function of the number of bins used to collect energy and given by:

${Loss} = {{10\;\log_{10}N^{2}} - {10\;{\log_{10}\left( {\sum\limits_{i = {{- M} - L}}^{{- M} + L}\left( \frac{\sin\left( {\pi\left\lbrack {M + {Nr}} \right\rbrack} \right)}{\sin\left( {\pi\left\lbrack {\frac{M + i}{N} + r} \right\rbrack} \right)} \right)^{2}} \right)}}}$

The worst case loss is experienced when the image falls midway between bins (r=1/(2N)). Using just one bin which is closest to the image results in a worst case loss of 3.9223[dB] using two bins results in a loss of 0.9120[dB] the following figure summarizes the loss as a function of the number of bins used.

We shall use 2 bins seems like a reasonable trade-off between complexity and performance.

FFT Processing of Probe2 (Single OFDM Symbol)

For simplicity consider a single OFDM symbol the extension to multi OFDM symbols will be given shortly after. We have shown that the FFT outputs at bins k and −k are given by

${Z\lbrack k\rbrack} = {\overset{\overset{{signal}\mspace{14mu}{term}}{︷}}{A_{1}{hK}_{1}} + \underset{\underset{{{ICI}\mspace{14mu}{from}\mspace{14mu}{bin}}\mspace{11mu} - k}{︸}}{B_{1}h^{*}K_{2}} + W_{k}}$ ${Z\left\lbrack {- k} \right\rbrack} = {\underset{\underset{{ICI}\mspace{14mu}{from}\mspace{14mu}{bin}\mspace{14mu} k}{︸}}{A_{2}{hK}_{1}} + \overset{\overset{{signal}\mspace{14mu}{term}}{︷}}{B_{2}h^{*}K_{2}} + W_{- k}}$

It is easy to show that each expression is composed of an expected signal term and an ICI term from the mirror frequency. We shall now show that the ICI terms are much smaller than the signal terms and can thus be neglected.

The ICI induced at bin k is due to the fact that the image signal that results from I/Q imbalance is produced at a digital frequency of

${- \frac{k}{N}} - {\varphi^{\prime}\left( {{{where}\mspace{14mu}\varphi^{\prime}} = \left( {\varphi - {\frac{1}{2}ɛ}} \right)} \right)}$ which is not on the FFT grid. The further away this frequency is from the FFT grid of

$\frac{l}{N}$ the larger the ICI. Since k is restricted to be in the interval {[146,186}[217,249]} the image is produced far away from the desired signal and the ICI noise it produces at frequency

$\frac{k}{N}$ is very small. To see this considers the ratio between the signal and ICI terms at bin k. We denote this ratio as the SNR between the desired and ICI terms and it is given by:

$\begin{matrix} {{SNR}_{k} = {10\;{\log_{10}\left( \frac{{{hNK}_{1}}^{2}}{{{h^{*}{\mathbb{e}}^{- {j{({2{\pi{({N - 1})}}{({\frac{k}{N} + \varphi^{\prime}})}})}}}\frac{\sin\left( {2{\pi\varphi}^{\prime}N} \right)}{\sin\left( {\frac{2\pi\; k}{N} + {2\pi\;\varphi^{\prime}}} \right)}K_{2}}}^{2}} \right)}}} \\ {= {{10\;{\log_{10}\left( \frac{N^{2}{K_{1}}^{2}}{{K_{2}}^{2}} \right)}} - {10\;{\log_{10}\left( \left\lbrack \frac{\sin\left( {2{\pi\varphi}^{\prime}N} \right)}{\sin\left( {\frac{2\pi\; k}{N} + {2\pi\;\varphi^{\prime}}} \right)} \right\rbrack^{2} \right)}}}} \end{matrix}$

The worst case SNR is found by minimizing the above expression with respect to {g, θ,k, φ′}. It is easy to show that minimizing the above expression is separable and thus minimization is achieved by

-   -   minimizing the first term with respect to g and θ under the         constraint that g         [0.5,2] (max 3[dB] amplitude imbalance) and θ         [−10°,10°]     -   maximizing the second term under the constraint that φ′         [−200e−6*1.5e9/50e6: −200e−6*1.5e9/50e6] and k         [146:186, 217:249]

The above minimizations were performed numerically using a Matlab simulation. FIG. 12 a depicts the first term as a function of g and θ

It is easy to see (analytically as well) that the minimum is at the edges of the argument interval namely for g=0.5,2 and Teta=±10° and thus

$57.4216 = {\min_{g,\theta}{\left\{ {10\;{\log_{10}\left( \frac{N^{2}{{K_{1}\left( {g,\theta} \right)}}^{2}}{{{K_{2}\left( {g,\theta} \right)}}^{2}} \right)}} \right\}\mspace{20mu}{st}\mspace{14mu}\begin{matrix} {g \in \left\lbrack {0.5,2} \right\rbrack} \\ {\theta \in \left\lbrack {{{- 10}{^\circ}},{10{^\circ}}} \right\rbrack} \end{matrix}}}$

FIG. 12 b depicts the second term as a function of φ and k

From FIG. 12 b it is easy to see that the second term is maximized for k=249, for such a k the second term is depicted in FIG. 13.

Maximum is achieved for Df=±245 Khz and thus

$17.04 = {\max_{g,\theta}\left\{ {10\;{\log_{10}\left( \left\lbrack \frac{\sin\left( {2{\pi\varphi}\; N} \right)}{\sin\left( {\frac{2\pi\; k}{N} + {2\pi\;\varphi}} \right)} \right\rbrack^{2} \right)}} \right\}}$ ${st}\mspace{14mu}\begin{matrix} {\varphi \in \left\lbrack {{{- \Delta}\; f},{\Delta\; f}} \right\rbrack} \\ {k \in {\left\lbrack {146,186} \right\rbrack\bigcup\left\lbrack {217,249} \right\rbrack}} \end{matrix}$

Thus the worst case SNR induced by the ICI term is 40.3816[dB]

${40.3816\mspace{14mu}\lbrack{dB}\rbrack} = {{\min\left( {SNR}_{k} \right)} = {\min_{\varphi,k,g,\theta}\left( {{10\;{\log_{10}\left( \frac{N^{2}{{K_{1}\left( {g,\theta} \right)}}^{2}}{{{K_{2}\left( {g,\theta} \right)}}^{2}} \right)}} - {10\;{\log_{10}\left( \left\lbrack \frac{\sin\left( {2{\pi\varphi}\; N} \right)}{\sin\left( {\frac{2\pi\; k}{N} + {2\pi\;\varphi}} \right)} \right\rbrack^{2} \right)}}} \right)}}$

Thus the ICI term is at the worst case 40 [dB] below the signal term and so can be neglected. A similar analysis can be performed for the negative bins. The FFT outputs at bins k and −k+i after neglecting the ICI terms is given by:

Z[k] = hAK₁ + W_(k) Z[−k + i] = h^(*)B_(i)K₂ + W_(−k + i) $A = {{\mathbb{e}}^{{j\pi ɛ}{({N - 1})}}\frac{\sin\left( {{\pi ɛ}\; N} \right)}{\sin({\pi ɛ})}}$ $B_{i} = {{\mathbb{e}}^{{- {{j2\pi}{({\varphi - {\frac{1}{2}ɛ} + \frac{i}{2N}})}}}{({N - 1})}}\left( \frac{\left( {- 1} \right)^{i}{\sin\left( {2\pi\;{N\left\lbrack {\varphi - {\frac{1}{2}ɛ}} \right\rbrack}} \right)}}{\sin\left( {2{\pi\left\lbrack {\varphi - {\frac{1}{2}ɛ} + \frac{i}{2N}} \right\rbrack}} \right)} \right)}$

Since we cannot estimate the channel response h we cannot solve a linear LS problem for ge^(−jθ), instead we first solve a LS problem for the estimation of hK*₂ from the two negative bins −k+i1 and −k+i2

$\begin{bmatrix} {{\hat{Z}}^{*}\left\lbrack {{- k} + i_{1}} \right\rbrack} \\ {{\hat{Z}}^{*}\left\lbrack {{- k} + i_{2}} \right\rbrack} \end{bmatrix} = {{\begin{bmatrix} B_{1}^{*} \\ B_{2}^{*} \end{bmatrix}\left( {hK}_{2}^{*} \right)} + \begin{bmatrix} W_{k + i_{1}}^{*} \\ W_{k + i_{2}}^{*} \end{bmatrix}}$ $\left( {hK}_{2}^{*} \right)_{LS} = \frac{{B_{1}{{\hat{Z}}^{*}\left\lbrack {{- k} + i_{1}} \right\rbrack}} + {B_{2}{{\hat{Z}}^{*}\left\lbrack {{- k} + i_{2}} \right\rbrack}}}{{B_{1}}^{2} + {B_{2}}^{2}}$

Thus we can estimate

$\frac{K_{1}}{K_{2}^{*}}$ without knowledge of the channel h by

${C \equiv \frac{{\overset{\_}{K}}_{1}}{K_{2}^{*}}} = {\frac{\left( {hK}_{1} \right)}{\left( {hK}_{2}^{*} \right)_{LS}} = {\frac{{Z\lbrack k\rbrack}/A}{{B_{1}{{\hat{Z}}^{*}\left\lbrack {{- k} + i_{1}} \right\rbrack}} + {B_{2}{{{\hat{Z}}^{*}\left\lbrack {{- k} + i_{2}} \right\rbrack}/{B_{1}}^{2}}} + {B_{2}}^{2}} = \frac{\left( {{B_{1}}^{2} + {B_{2}}^{2}} \right) \cdot {Z\lbrack k\rbrack}}{A \cdot \left( {{B_{1}{{\hat{Z}}^{*}\left\lbrack {{- k} + i_{1}} \right\rbrack}} + {B_{2}{{\hat{Z}}^{*}\left\lbrack {{- k} + i_{2}} \right\rbrack}}} \right)}}}$

Since probe2 is composed of two tones one at k1 and the other at k2 we can average the result from these two tones and thus

$\overset{\_}{C} \equiv {{\frac{1}{2}\frac{\left( {{B_{1}}^{2} + {B_{2}}^{2}} \right) \cdot {Z\left\lbrack k_{1} \right\rbrack}}{A \cdot \left( {{B_{1}{{\hat{Z}}^{*}\left\lbrack {{- k_{1}} + i_{1}} \right\rbrack}} + {B_{2}{{\hat{Z}}^{*}\left\lbrack {{- k_{1}} + i_{2}} \right\rbrack}}} \right)}} + {\frac{1}{2}\frac{\left( {{B_{1}}^{2} + {B_{2}}^{2}} \right) \cdot {Z\left\lbrack k_{2} \right\rbrack}}{A \cdot \left( {{B_{1}{{\hat{Z}}^{*}\left\lbrack {{- k_{2}} + i_{1}} \right\rbrack}} + {B_{2}{{\hat{Z}}^{*}\left\lbrack {{- k_{2}} + i_{2}} \right\rbrack}}} \right)}}}$

It is easy to see that

${g\;{\mathbb{e}}^{- {j\theta}}} = \frac{{K_{1}/K_{2}^{*}} - 1}{{K_{1}/K_{2}^{*}} + 1}$

And thus its estimate is given by

$\overset{\_}{g\;{\mathbb{e}}^{- {j\theta}}} = \frac{\overset{\_}{C} - 1}{\overset{\_}{C} + 1}$

The I/Q compensation is then easily computed by

$\hat{\xi} = \frac{1}{{real}\left\{ \overset{\_}{g\;{\mathbb{e}}^{- {j\theta}}} \right\}}$ $\hat{\rho} = {- \frac{{imag}\left\{ \overset{\_}{g\;{\mathbb{e}}^{- {j\theta}}} \right\}}{{real}\left\{ \overset{\_}{g\;{\mathbb{e}}^{- {j\theta}}} \right\}}}$ FFT Processing of Probe2 (Multi OFDM Symbol)

When looking at multiple OFDM symbols we need to take into account the phase error induced by the accumulation of the residual frequency error ε. It is easy to show that the phase of the m'th OFDM symbol relative to the first one is given by

${Z\left\lbrack {k \cdot m} \right\rbrack} = \left\{ \begin{matrix} {k \geq 0} & {{\mathbb{e}}^{{{j2\pi ɛ}{({N + N_{CP}})}}m}{Z\left\lbrack {k,0} \right\rbrack}} \\ {k < 0} & {{\mathbb{e}}^{{{j2\pi}{({{2\varphi} - ɛ})}}{({N + N_{CP}})}m}{Z\left\lbrack {k,0} \right\rbrack}} \end{matrix}\quad \right.$

Note that the above does not take into account the sampling frequency error its effect is assumed to be small and was neglected throughout the analysis.

The phase accumulated from the starting time of carrier frequency compensation to the start of the first FFT window should be accounted for. Since our algorithm computed the ratio between Zk and conj(Z−k) any constant phase term will not cancel out but on the contrary double itself.

Thus the FFT output at bins +k, −k+i for the m'th OFDM symbol is given by

${Z\lbrack k\rbrack} = {{\underset{\underset{\# 1}{︸}}{D}\overset{\overset{\# 2}{︷}}{{\mathbb{e}}^{{- {j2\pi\Delta}}\;{n{({\varphi - ɛ})}}}}\underset{\underset{\# 3}{︸}}{{\mathbb{e}}^{{{j2\pi ɛ}{({N + N_{CP}})}}m}}{hAK}_{1}} + W_{k}}$ ${Z\left\lbrack {{- k} + i} \right\rbrack} = {{\underset{\underset{\# 1}{︸}}{D^{*}}\overset{\overset{\# 2}{︷}}{{\mathbb{e}}^{{- {j2\pi\Delta}}\;{n{({\varphi - ɛ})}}}}\underset{\underset{\# 3}{︸}}{{\mathbb{e}}^{{{j2\pi}{({2 - {\varphi ɛ}})}}{({N + N_{CP}})}m}}h^{*}B_{i}K_{2}} + W_{{- k} + i}}$ $\begin{matrix} {{\# 1}\text{-}\mspace{11mu}{due}\mspace{14mu}{to}\mspace{14mu}\Delta\; n\mspace{14mu}{samples}\mspace{14mu}{between}\mspace{14mu}{start}\mspace{14mu}{of}\mspace{14mu}{compensation}\mspace{14mu}{to}\mspace{14mu}{FFT}} \\ {\mspace{45mu}{{will}\mspace{14mu}{not}\mspace{14mu}{cancell}\mspace{14mu}{out}}} \\ {{\# 2}\text{-}\mspace{11mu}{due}\mspace{14mu}{to}\mspace{14mu}\Delta\; n\mspace{14mu}{samples}\mspace{14mu}{between}\mspace{14mu}{start}\mspace{14mu}{of}\mspace{14mu}{compensation}\mspace{14mu}{to}\mspace{14mu}{FFT}} \\ {\mspace{45mu}{{start}\mspace{14mu}{will}\mspace{14mu}{not}\mspace{14mu}{cancell}\mspace{14mu}{out}}} \\ {{\# 3}\text{-}\mspace{11mu}{Phase}\mspace{14mu}{due}\mspace{14mu}{to}\mspace{14mu}{residual}\mspace{14mu}{carrier}\mspace{14mu}{freq}\mspace{14mu}{error}} \end{matrix}$ Residual Carrier Frequency Estimation

To use the information from all L OFDM symbols we need to compensate for the residual frequency offset ε and then compute the average of the compensated signals form each bin to reduce the AWGN variance. Since ε can be large enough such that phase wrapping can occur several times during the L OFDM symbols we propose the following estimator which is immune to phase wrapping (as long as no more than one wrap occurs between two consecutive samples which is the case here).

Residual Frequency Estimation

The residual frequency error estimate may be computed by

$\hat{ɛ} = \frac{{angle}\left( {\sum\limits_{m = o}^{L - 2}{{Z\left\lbrack {k_{i},m} \right\rbrack}{Z^{*}\left\lbrack {k_{i},{m - 1}} \right\rbrack}}} \right)}{2{\pi\left( {N + N_{CP}} \right)}}$ Where $k_{i} = \left\{ \begin{matrix} k_{1} & {{SNR}_{k_{1}} > {SNR}_{k_{2}}} \\ k_{2} & {{SNR}_{k_{2}} > {SNR}_{k_{1}}} \end{matrix} \right.$

The residual frequency error compensation and averaging is given by Residual Frequency Compensation and time averaging

${\overset{\_}{Z}}_{k_{1}} = {\sum\limits_{m = 0}^{L - 1}{{Z\left\lbrack {k_{1},m} \right\rbrack}{\mathbb{e}}^{{- {j2\pi}}\;{\hat{ɛ}{({N + N_{CP}})}}m}}}$ ${\overset{\_}{Z}}_{k_{2}} = {\sum\limits_{m = 0}^{L - 1}{{Z\left\lbrack {k_{2},m} \right\rbrack}{\mathbb{e}}^{{- {j2\pi}}\;{\hat{ɛ}{({N + N_{CP}})}}m}}}$ ${\overset{\_}{Z}}_{{- k_{1}} + i_{1}} = {\sum\limits_{m = 0}^{L - 1}{{Z\left\lbrack {{{- k_{1}} + i_{1}},m} \right\rbrack}{\mathbb{e}}^{{- {{j2\pi}{({{2\;\hat{\varphi}} + \hat{ɛ}})}}}{({N + N_{CP}})}m}}}$ ${\overset{\_}{Z}}_{{- k_{2}} + i_{1}} = {\sum\limits_{m = 0}^{L - 1}{{Z\left\lbrack {{{- k_{2}} + i_{1}},m} \right\rbrack}{\mathbb{e}}^{{- {{j2\pi}{({{2\;\hat{\varphi}} + \hat{ɛ}})}}}{({N + N_{CP}})}m}}}$ ${\overset{\_}{Z}}_{{- k_{1}} + i_{2}} = {\sum\limits_{m = 0}^{L - 1}{{Z\left\lbrack {{{- k_{1}} + i_{2}},m} \right\rbrack}{\mathbb{e}}^{{- {{j2\pi}{({{2\;\hat{\varphi}} + \hat{ɛ}})}}}{({N + N_{CP}})}m}}}$ ${\overset{\_}{Z}}_{{- k_{2}} + i_{2}} = {\sum\limits_{m = 0}^{L - 1}{{Z\left\lbrack {{{- k_{2}} + i_{2}},m} \right\rbrack}{\mathbb{e}}^{{- {{j2\pi}{({{2\;\hat{\varphi}} + \hat{ɛ}})}}}{({N + N_{CP}})}m}}}$

The phasor ge^(−jθ) can then be estimated using the same estimator derived above, namely

$C_{i} = {{\mathbb{e}}^{{{j4\pi}{({\varphi - ɛ})}}{({\Delta\; n})}}\frac{\left( {{B_{1}}^{2} + {B_{2}}^{2}} \right) \cdot {\overset{\_}{Z}}_{k_{i}}}{{B_{1}\left( {\overset{\_}{Z}}_{{- k_{i}} + i_{1}} \right)}^{*} + {B_{2}\left( {\overset{\_}{Z}}_{{- k_{i}} + i_{2}} \right)}^{*}}}$

Where the phase term e^(j4π(φ−ε)(Δn)) compensates for the initial phase error accumulated from the start time of frequency compensation till the start of the first FFT window.

Simplification of the coefficients Bi and A

For pragmatic implementation we need to simplify the expressions of Bi and A, simplification can be obtained by introducing some approximations. Let's look at

$A = {{\mathbb{e}}^{{j\pi ɛ}{({N - 1})}}\frac{\sin\left( {{\pi ɛ}\; N} \right)}{\sin\left( {\pi\; ɛ} \right)}}$

The residual frequency error is typically smaller then 10 khz (7 ppm) for such an error

𝕖^(jπɛ(N − 1)) = 0.9872 + 0.1595i ≅ 1 $\frac{\sin\left( {{\pi ɛ}\; N} \right)}{\sin\left( {\pi\; ɛ} \right)} = {{254.8975 \cong 256} = N}$

Thus we make the following approximation

$A = {{{\mathbb{e}}^{{j\pi ɛ}{({N - 1})}}\frac{\sin\left( {{\pi ɛ}\; N} \right)}{\sin\left( {\pi\; ɛ} \right)}} \cong N}$

As for Bi

$\begin{matrix} \begin{matrix} {B_{i} = {{\mathbb{e}}^{{- {{j2\pi}{({\varphi - {\frac{1}{2}ɛ} + \frac{\mathbb{i}}{2N}})}}}{({N - 1})}}\left( \frac{\left( {- 1} \right)^{i}{\sin\left( {2\pi\;{N\left\lbrack {\varphi - {\frac{1}{2}ɛ}} \right\rbrack}} \right)}}{\sin\left( {2{\pi\left\lbrack {\varphi - {\frac{1}{2}ɛ} + \frac{\mathbb{i}}{2N}} \right\rbrack}} \right)} \right)}} \\ {\cong {{\mathbb{e}}^{{- {{j2\pi}{({\varphi - {\frac{1}{2}ɛ}})}}}{(N)}}{{\mathbb{e}}^{- {j\pi\mathbb{i}}}\left( \frac{\left( {- 1} \right)^{i}{\sin\left( {2\pi\;{N\left\lbrack {\varphi - {\frac{1}{2}ɛ}} \right\rbrack}} \right)}}{\sin\left( {2{\pi\left\lbrack {\varphi - {\frac{1}{2}ɛ} + \frac{\mathbb{i}}{2N}} \right\rbrack}} \right)} \right)}}} \\ {\cong {{{\mathbb{e}}^{{- {j2\pi}}\;{N{({\varphi - ɛ})}}}\left( {- 1} \right)}^{2i}\left( \frac{\sin\left( {2\pi\;{N\left\lbrack {\varphi - ɛ} \right\rbrack}} \right)}{\sin\left( {2{\pi\left\lbrack {\varphi - ɛ + \frac{\mathbb{i}}{2N}} \right\rbrack}} \right)} \right)}} \\ {= {{\mathbb{e}}^{{- {j2\pi}}\;{N{({\varphi - ɛ})}}}\left( \frac{\sin\left( {2\pi\;{N\left\lbrack {\varphi - ɛ} \right\rbrack}} \right)}{\sin\left( {2{\pi\left\lbrack {\varphi - ɛ + \frac{\mathbb{i}}{2N}} \right\rbrack}} \right)} \right)}} \end{matrix} \\ {{{{Where}\mspace{14mu}\varphi} - ɛ} = \frac{CFO}{2\pi}} \\ \begin{matrix} {B_{i} \cong {{\mathbb{e}}^{{- {j2\pi}}\;{N{({\varphi - ɛ})}}}\left( \frac{\sin\left( {2\pi\;{N\left\lbrack {\varphi - ɛ} \right\rbrack}} \right)}{\sin\left( {2{\pi\left\lbrack {\varphi - ɛ + \frac{\mathbb{i}}{2N}} \right\rbrack}} \right)} \right)}} \\ {= \frac{{\sin\left( {2 \cdot N \cdot {CFO}} \right)} + {j\left\lbrack {{\cos\left( {2 \cdot N \cdot {CFO}} \right)} - 1} \right\rbrack}}{2{\sin\left( {2{\pi\left\lbrack {\varphi - ɛ + \frac{\mathbb{i}}{2N}} \right\rbrack}} \right)}}} \end{matrix} \end{matrix}$

Since the frequency shift along with the residual frequency error is smaller than (200+7)ppm and −3≦i≦3 follows that the argument of the sin( ) in the denominator is small

${{2{\pi\left\lbrack {\varphi - ɛ + \frac{\mathbb{i}}{2N}} \right\rbrack}} < {{207e} - {6*1.5e\; 9\text{/}50e\; 6} + {3\text{/}512}}} = 0.0121$

For such a small angle a simple linear approximation has very little error

$\begin{matrix} {{\sin(0.0121)} \cong 0.0121} \\  \Downarrow \\ {{\sin\left( {2{\pi\left\lbrack {\varphi - ɛ + \frac{\mathbb{i}}{2N}} \right\rbrack}} \right)} \cong {2{\pi\left\lbrack {\varphi - ɛ + \frac{\mathbb{i}}{2N}} \right\rbrack}}} \end{matrix}$

Thus follows that

$\begin{matrix} {B_{i} \cong \frac{{\sin\left( {2 \cdot N \cdot {CFO}} \right)} + {j\left\lbrack {{\cos\left( {2 \cdot N \cdot {CFO}} \right)} - 1} \right\rbrack}}{2{\sin\left( {2{\pi\left\lbrack {\varphi - ɛ + \frac{\mathbb{i}}{2N}} \right\rbrack}} \right)}}} \\ {\cong {\frac{1}{2}\frac{{\sin\left( {2 \cdot N \cdot {CFO}} \right)} + {j\left\lbrack {{\cos\left( {2 \cdot N \cdot {CFO}} \right)} - 1} \right\rbrack}}{{CFO} + \frac{\mathbb{i}\pi}{N}}}} \end{matrix}$

And so the simplified coefficients are given by

A = N $B_{1} = {{\frac{1}{2}\left( \frac{\sin\left( {2 \cdot {CFO} \cdot N} \right)}{{CFO} + \frac{\pi \cdot i_{1}}{N}} \right)} + {\frac{j}{2}\left( \frac{{\sin\left( {2 \cdot {CFO} \cdot N} \right)} - 1}{{CFO} + \frac{\pi \cdot i_{1}}{N}} \right)}}$ $B_{2} = {{\frac{1}{2}\left( \frac{\sin\left( {2 \cdot {CFO} \cdot N} \right)}{{CFO} + \frac{\pi \cdot i_{2}}{N}} \right)} + {\frac{j}{2}\left( \frac{{\sin\left( {2 \cdot {CFO} \cdot N} \right)} - 1}{{CFO} + \frac{\pi \cdot i_{2}}{N}} \right)}}$

APPENDIX B

Exemplary Measurements

Ideal Channel No AWGN with 200 ppm carrier & sampling frequency offset

The following plots summarize simulation results for a 3 [dB] amplitude imbalance, 10° phase imbalance, 200 ppm frequency offset, ideal channel and no AWGN. Before RX I/Q compensation routine is invoked the receiver SNR is around 10.1 [dB] as can be seen in the following FIG. B-1.

FIG. 14 shows a Slicer Input Ideal Channel, No AWGN Before Cancellation

After processing of the first probe2, the SNR is around 33 [dB]. In the following FIG. B-2 one can compare the image signal magnitude before and after the first iteration.

FIG. B-2: Frequency Input Ideal Channel, No AWGN after First Iteration

After processing of the second Probe II, the SNR is around 39.6[dB]. In following FIG. B-3, one can compare the image signal magnitude before and after the second iteration. The image signal is no longer visible after the second iteration.

FIG. B-3: Frequency Plot Ideal Channel No AWGN after Two Iterations

After processing of the third Probe II, the SNR is around 40.6[dB].

FIG. B-4: Slicer Input Ideal Channel, No AWGN after Three Iterations

The following FIG. B-5 depicts the slicer SNR after processing of the fourth Probe II transmission SNR is shown to be around 40.9[dB]

FIG. B-5: Slicer Input Ideal Channel, No AWGN after Four Iterations

The following FIG. B-6 depicts the slicing SNR without I/Q imbalance, the SNR is around 41.3[dB]. Thus comparing this SNR to that of the SNR obtained after four probeII transmission we can conclude that the residual I/Q imbalance degrades performance by around 0.4[dB] relative to a noise floor of 41.3 [dB].

FIG. B-6: Slicer Input Channel, No AWGN no I/Q Imbalance

The I/Q imbalance parameter estimation after each one of the four iterations is summarized in the following table

Iteration Number Gain Theta [°] Slicer SNR [dB] True Value 0.70795 10 41.3 #0 1 0 10.1 #1 0.71345 11.4402 33 #2 0.71103 10.3227 39.6 #3 0.70989 10.1742 40.6 #4 0.70944 10.1358 40.9

Channel MoCA10408, SNR AWGN 15 [dB]

The following plots summarize simulation results for a 3 [dB] amplitude imbalance, 10° phase imbalance, 200 ppm frequency offset, MoCA10408 channel and 15 [dB] AWGN SNR. Before RX I/Q compensation routine is invoked the receiver SNR is around 5.1 [dB] as can be seen in

FIG. B-7: Slicer Input MoCA10408 Channel, 15 [dB] AWGN SNR Before Cancellation

After processing of the first Probe II, the SNR is around 9.5 [dB]. In the following FIG. B-8 one can compare the image signal magnitude before and after the first iteration.

FIG. B-8: Frequency Plot MoCA10408 Channel 15 [dB] AWGN after First Iteration

FIG. B-9: Slicer Input MoCA10408 Channel 15 [dB] AWGN SNR after First Iteration.

After processing of the second Probe II, the SNR is around 11 [dB].

FIG. B-10: Frequency Plot MoCA10408 Channel 15 [dB] after Second Iteration

FIG. B-11: Slicer Input MoCA10408 Channel 15 [dB] AWGN SNR after Second Iteration.

After processing of the third Probe II, the SNR is around 11.6[dB]

FIG. B-12: Frequency Plot MoCA10408 Channel 15 [dB] AWGN SNR after Third Iteration

FIG. B-13: Slicer Input MoCA10408 Channel 15 [dB] AWGN SNR after Second Iteration

The SNR when no I/Q imbalance is present at the receiver is around 11.3 [dB] thus the residual I/Q imbalance is well below the noise floor of our demodulator and the estimation and compensation algorithm is robust even under harsh channel conditions.

FIG. B-14: Slicer Input MoCA10408 Channel 15 [dB] AWGN SNR no I/Q Imbalance

Decoupling of TX and RX I/Q Imbalance

The intentional frequency shift specified by MoCA results in the decoupling of the TX and RX imbalance parameters, to show that our algorithm can estimate the RX parameters in the presence of X imbalance we show simulation results for the following scenario

-   -   TX amplitude imbalance 1 [dB]     -   TX phase imbalance 2°     -   RX amplitude imbalance 3 [dB]     -   RX phase imbalance 10°     -   Frequency Offset 20 ppm     -   Channel=Ideal, no AWGN

Before transmission of Probe2 the SNR was around 11.3 [dB]

FIG. B-15: Slicer Input Ideal Channel No AWGN, Under RX and TX I/Q Imbalance

After three probe2 transmissions the SNR was around 21.2 [dB].

FIG. B-16: Frequency Plot Idea Channel No AWGN, Under RX and TX I/Q Imbalance after Third Iteration

FIG. B-17: Slicer Input Ideal Channel No AWGN, Under RX and TX I/Q Imbalance after Third Iteration

The estimated RX I/Q imbalance parameters after 3 iterations were

Parameter True Estimated G 0.7071 0.70785 ⊖ 10.0 10.2014

Thus parameters were correctly estimated, for comparison the SNR in a scenario where only TX imbalance is present is around 21.2 [dB].

FIG. 15 shows a Slicer Input Ideal Channel No AWGN Under TX I/Q Imbalance Only

Thus the proposed algorithm is robust in the presence of TX I/Q imbalance.

APPENDIX C

Fixed Point Pseudo Code

The following flow Pseudo code gives a fixed point implementation of the above algorithm. Note that complex variables have the letter “c” prepended.

Function 1: Probe2Processing

function [rho, theta, Scale_Q] = Probe2Processing (CFO, cIQparameters_log,i1,i2, N_delta) if (CFO==0)   CFO=−1; end % Residual Frequency Estimation [cphasor_p,cphasor_m] = Residual_Frequency_Estimation(cIQparameters_log,CFO); % Residual Frequency Correction cZ = Residual_frequency_Correction(cIQparameters_log,cphasor_p,cphasor_m); % Coefficient Computation [cfB1,cfB2,Scale_ratio] = Coeff_Computation(CFO,i1,i2); % Phasor Estimation [cg_exp_mTeta_M_16,Scale_g,scale_inv] = Phasor_Estimation (cZ,cfB1,cfB2,CFO,N_delta,Scale_ratio); % IQ Coeff Computation [theta,rho, Scale_Q] = Compensation_Params_Estimation(cg_exp_mTeta_M_16, scale_inv,Scale_g);

TABLE C-1 Probe2Processing Variable Definition Table Name Size Comments I/O CFO 32 bit Carrier Frequency Offset I cIQparameters_log 16 bit × 6 × Nsym Complex Recorded FFT outputs I I1  8 bit Negative Bin offset 1 I I2  8 bit Negative Bin offset 2 I N_delta 16 bit NCO reset time offset I cphasor_p 16 bit Complex Frequency compensation NA cphasor_m 16 bit Complex Frequency compensation NA cZ 16 bit × 6 Complex Rotated FFT output NA cfB1 16 bit Complex Estimation Coefficient NA cfB2 16 bit Complex Estimation Coefficient NA Scale_ratio 16 bit Scaling factor NA Cg_exp_mTeta_M_16 16 bit Complex Estimated I/Q Phasor NA Theta 16 bit I/Q Compensation O Rho 16 bit I/Q Compensation O Function 2: Residual_Frequency_Estimation

function [cphasor_p,cphasor_m] = Residual_Frequency_Estimation(cIQparameters_log,CFO) %TBD Select 1 or 2 according to SNR cfphasor_64 = 0; for i=0:Nsym−2  cfphasor_64 = Cmplx_Add_64_32(cfphasor_64, Cmplx_Mult_16_16(cIQparameters_log(i,1),...                 conj(cIQparameters_log(i+1,1)); end %Level control to 16 bit signed Nphasor_bits = Nfft_out−1; csphasor = Scale_Complex_64(cfphasor_64, Nphasor_bits); %Get phasor angle and magnitude [angle_Ef rPhasor] = cordic_SW( csphasor,1); %Generate ‘Exp_Vec_p’ fScale = 26981;      %const 16 bit − round(gcordic)*2{circumflex over ( )}(Nfft_out−1)) csphasor_div=Cmplx_real_div_32_16 (csphasor<<(Nfft_out−2)), rPhasor); %32bit complex / 16bit real  division % csphasor_div <+−2{circumflex over ( )}15 cphasor_p_32 = (Cmplx_real_mult_16_16(fScale,csphasor_div) )>>(Nfft_out−2); %Scale back to 16bit,  known cphasor_p = Cmplx_Saturate(cphasor_p_32, Nphasor_bits); %Compute angle for IQ image rotation (Coridic Preparations) angle_m = 2*CFO*(Nfft+LCP) − angle_Ef;   %1rad= 2{circumflex over ( )}(Fr_bits−1) %Generate ‘Exp_Vec_m’ [cphasor_m tmp] = cordic_SW( angle_m,0); %gives -phasor(angle) Function 3: Scale_Complex_(—)64

function [csphasor] = Scale_Complex_64(cfphasor_64, Nphasor_bits) Ceil_Log2_Abs_Real_Cfphasor_64 = ceil_log2(abs(real(cfphasor_64))); Ceil_Log2_Abs_Imag_Cfphasor_64 = ceil_log2(abs(imag(cfphasor_64))); Scale=Nphasor_bits-max(Ceil_Log2_Abs_Real_Cfphasor_64, Ceil_Log2_Abs_Imag_Cfphasor_64); if (Scale>=0) csphasor_32 = (cfphasor_64<< Scale); else csphasor_32 = (cfphasor_64>> (−Scale)); end csphasor = Cmplx_Saturate(csphasor_32, Nphasor_bits); Function 4: ceil_log 2

function [i] = ceil_log2 (X) i=0; while(X!=0)  X=X>>1;  i=i+1; end Function: 5: Sign

function [Y] = Sign (X) y=1; if (X<0)  y=−1; end Function 6: Cmplx_Saturate

function [X_16] = Cmplx_Saturate(X_32,Nbits) Sign_Real_X_32 = sign(real(X_32)); Sign_Imag_X_32 = sign(imag(X_32)); Abs_Real_X_32 = abs(real(X_32)); Abs_Imag_X_32 = abs(imag(X_32)); if (Abs_Real_X_32 >= (1<< Nbits) )  if(Sign_Real_X_32==1)   X_32.r= (1<< Nbits)−1;  else X_32.r= −( (1<< Nbits)−1 );  end end if (Abs_Imag_X_32 >= (1<< Nbits) )  if(Sign_Image_X_32==1)   X_32.i= (1<< Nbits)−1;  else X_32.i= −( (1<< Nbits)−1 );  end end X_16 = X_32; %casting to 16bit

TABLE C-2 Residual_Frequency_Estimation, Scale_Complex_64 Variable Definition Table Name Size Comments I/O cfphasor64 64 bit complex Phasor Acc NA Nphasor_bits  8 bit Target Num of phasor bits NA csphasor_32 32 bit complex Tmp Variable NA csphasor 16 bit complex Scaled phasor for freq rot NA rPhasor 16 bit Magnitude csphasor NA csphasor_div 16 bit complex Normalized phasor NA cphasor_p 16 bit complex Frequency Compensation O angle_Ef 32 bit Angle of csphasor NA angle_m 32 bit Angle for phasor_m rotation NA cphasor_m 16 bit complex Frequency Compensation O Function 7: Cordic_SW

function [z,x] = cordic_SW(data_in ,mode) %Inits & constants QUARTER= 102944;  %round( pi/2*2{circumflex over ( )}(Fr_bits−1)); Niter=13; %Do not Change without Zak's permission! Ntan=16; x_tmp(0)=0;x_tmp(1)=0; Atan_Table = [51472,30386,16055,8150,4091,2047,1024,512,256,128,64,32,16]; if(mode==0)  %map phase into [−pi/2 pi/2] interval and init Cordic  [Angle x] = Cordic_Pre_Process(data_in,0);  %Initial direction of rotation  if(Angle==0)    Sgn =1;  else   Sgn = −sign(Angle);  end else  %init Cordic  Angle=0;  %map phasor into [−pi/2 pi/2] interval and init Cordic  [data_in Angle_Offset] = Cordic_Pre_Process(data_in,1);  x(0) = real(data_in);  x(1) = imag(data_in);  if(x(1) ==0)   Sgn = 1;  else   Sgn = sign(x(1));  end end

TABLE C-3 SW CORDIC variable definition Name Size Comments I/O x_tmp 32 bit × 2 array Local memory NA (no more than 17 bits used) Atan_Table 16 bit × 13 array ATan Table (const) NA data_in 32 bit complex CORDIC input I Angle 32 bit Phasor Angle NA Angle_offset 32 bit Phasor Angle offset NA X 32 bit × 2 array CORDIC out 1 O z 32 bit × 2 array CORDIC out 2 O mode Boolean (1 bit) CORDIC mode select I Function 8: Cordic_Pre_Process

function [data_in, X] = Cordic_Pre_Process(data_in,mode) %*********************************************** % Pre Cordic Processing Bring angle to [−pi/2,pi/2] and find Quadrate %*********************************************** if(mode==0)  Npi = 0;  Sgn=0;  while (data_in < −QUARTER)   data_in = data_in + (QUARTER<<1);   Npi = 1−Npi;  end  while (data_in > QUARTER)   data_in = data_in − (QUARTER<<1);   Npi=1−Npi;  end  X(0) = 19898;           % (1/GainCordic)*2{circumflex over ( )}(Ntan−1)  X(1) = 0;  if(Npi)   X=−X;  end else  X=0;  Sgn_real = sign(real(data_in));  Sgn_imag = sign(imag(data_in));  if ( Sgn_real ==−1)   data_in = −data_in;   if(Sgn_imag == 1)    X = QUARTER<<1;   else    X = −(QUARTER<<1);   end  end end

TABLE C-4 Cordic_SW, Cordic_Pre_Process variable definition Name Size Comments I/O data_in 32 bit complex CORDIC input I/O X 16 bit × 2 array CORDIC Phasor O Function 9: Residual_Frequency_Compensation

function [cZ_16] = Residual_frequency_Correction(cIQparameters_log,cphasor_p,cphasor_m) cdphasor_p = cphasor_p; cdphasor_m = cphasor_m; cZ_ACC_64 = zeros(1,6);%64bit Acc array (16bit phasor * 16bit FFT output + log2(40)bit for acc >32   !!!!) for i=1:Nsym−1   cZ_ACC_64(1) = Cmplx_Add_64_32(cZ_ACC_64(1), Cmplx_Mult_16_16(cIQparameters_log(i+1,1)   ,cphasor_p);  cZ_ACC_64(2) = Cmplx_Add_64_32(cZ_ACC_64(2), Cmplx_Mult_16_16(cIQparameters_log(i+1,2)   ,cphasor_p);  cZ_ACC_64(3) = Cmplx_Add_64_32(cZ_ACC_64(3), Cmplx_Mult_16_16(cIQparameters_log(i+1,3)   ,cphasor_m);  cZ_ACC_64(4) = Cmplx_Add_64_32(cZ_ACC_64(4), Cmplx_Mult_16_16(cIQparameters_log(i+1,4)   ,cphasor_m);  cZ_ACC_64(5) = Cmplx_Add_64_32(cZ_ACC_64(5), Cmplx_Mult_16_16(cIQparameters_log(i+1,5)   ,cphasor_m);  cZ_ACC_64(6) = Cmplx_Add_64_32(cZ_ACC_64(6), Cmplx_Mult_16_16(cIQparameters_log(i+1,6)   ,cphasor_m);  cphasor_p = Cmplx_Saturate ((Cmplx_Mult_16_16 (cphasor_p,cdphasor_p))>>(Ntan−1), Ntan−1);  cphasor_m = Cmplx_Saturate( (Cmplx_Mult_16_16 (cphasor_m.cdphasor_m))>>(Ntan−1), Ntan−1); end %Scale Back to fit 32bit for i=1:6 cZ_ACC_32(i) = cZ_ACC_64(i)>>(Nfft_out); end % Level control vector so that max fits in 16bit Max_Z=0; for i=1:6  if (abs(real(cZ_ACC_32(i)))>Max_Z)   Max_Z = abs(real(Z_ACC_32(i)));  end  if (abs(imag(cZ_ACC_32(i)))>Max_Z)   Max_Z = abs(imag(Z_ACC_32(i)));  end end %Scale Back to 16 bits Scale_Z = (Nfft_out−1) − ceil_log2(Max_Z); for i=1:6 cZ_16(i) = Cmplx_Saturate (cZ_ACC_32(i)>>(Scale_Z), Nfft_out−1); end

TABLE C-5 Residual_frequency_Compensation variable Name Size Comments I/O cdphasor_p 16 bit Complex NA cdphasor_m 16 bit Complex NA cphasor_p 16 bit Complex I cphasor_m 16 bit Complex I cZ_ACC_64 64 bit Complex × 6 array NA cZ_32 32 bit Complex × 6 array NA cZ_16 16 bit Complex × 6 array O Max_32 32 bit NA ScaleZ 16 bit NA Function: 10: Coeff Computation

function [cfB1,cfB2,Scale_ratio] = Coeff_Computation(CFO,i1,i2) Log2Nfft = 8; %log2(Nfft) = log2(256)=8; PI_over_FFT = 804; % round(pi/Nfft*2{circumflex over ( )}(Fr_bits−1)) Nfft=256; N1=2; [cphasor11 tmp] = cordic_SW( (CFO<<(1+Log2Nfft)),0); cphasor11 = Switch_real_imag(cphasor11); cfs11 = Cmplx_Add_16_16 (phasor11, −sqrt(−1)*(1<<(Ntan−1))); fs21 = CFO+ PI_over_FFT *i1; fs22 = CFO +PI_over_FFT *i2; if(fs21==0)  cfB1_32 = 1<<( Nfft_out+ Log2Nfft ); else  cfB1_32 = Cmplx_real_div_32_16 ((cfs11<<(Nfft_out−N1)),fs21); end if(fs22==0)  cfB2_32 = 1<<( Nfft_out+ Log2Nfft ); else  cfB2_32 = Cmplx_real_div_32_16 ((cfs11<<(Nfft_out−N1)),fs22); end %Scale to fB1,fB2 to 15bit signed Max =0; Max_abs_real_cfB1 = abs(real(cfB1_32)); Max_abs_imag_cfB1= abs(imag(cfB1_32)); Max_abs_real_cfB2 = abs(real(cfB2_32)); Max_abs_imag_cfB2= abs(imag(cfB2_32)); if(Max_abs_real_cfB1 >Max)  Max = Max_abs_real_cfB1; end if(abs Max_abs_imag_cfB1>Max)  Max = Max_abs_imag_cfB1; end if(Max_abs_real_cfB2 >Max)  Max = Max_abs_real_cfB2 end if(Max_abs_imag_cfB2>Max)  Max Max_abs_imag_cfB2;  end  Scale_ratio = 15−(ceil_log2(Max)+1); If (Scale>=0)  cfB1 = Cmplx_Saturate( (cfB1_32<<Scale_ratio), Nfft_out−2);  cfB2 = Cmplx_Saturate ((cfB2_32<<Scale_ratio),Nfft_out−2); else  cfB1 = Cmplx_Saturate ((cfB1_32>>−Scale_ratio), Nfft_out−2);  cfB2 = Cmplx_Saturate ((cfB2_32>>−Scale_ratio), Nfft_out−2); end

TABLE C-6 Coeff Computation Variable Definition Name Size Comments I/O cphasor11 16 bit complex CORDIC out NA cfs11 32 bit complex Coeff Numerator NA fs21 16 bit Coeff Denominator NA fs22 16 bit Coeff Denominator NA cfB1_32 32 bit complex Unscaled Coeff 1 NA cfB2_32 32 bit complex Unscaled Coeff 2 NA cfB1 16 bit complex Coeff 1 O cfB2 16 bit complex Coeff 2 O Scale_ratio 16 bit Coeff Scale factor O Function 11: Phasor Estimation Variable Definition

function [cg_exp_mTeta_M_16,Scale_g,scale_inv] = Phasor_Estimation(cZ,cfB1,cfB2,CFO,N_delta,Scale_ratio) N2=2; %**************************************************************** %  Fix Point Computation of % %  Z1*( |fB1|{circumflex over ( )}2+|fB2|{circumflex over ( )}2)  Z1*( |fB1|{circumflex over ( )}2+|fB2|{circumflex over ( )}2)*(B1‘*Z3 +B2*Z4’)’ % -------------------------- = --------------------------------------------- %   B1‘*Z3 +B2*Z4’      |B1‘*Z3|{circumflex over ( )}2 +|B2*Z4’|{circumflex over ( )}2 %**************************************************************** sumfB1SfB2S = (Real_Add_30_30_T(MAG_2_16(cfB1), MAG_2_16(cfB2)))>>15 cNUMERATOR_32(0) = Cmplx_Real_Mul_16_16(cZ(0),sumfB1SfB2S); cNUMERATOR_32(1) = Cmplx_Real_Mul_16_16(cZ(1),sumfB1SfB2S); cDENOM_32(0) =Cmplx_Add_32_32(Cmplx_Mul_16_16(conj(cZ(2)),(cfB1)),           Cmplx_Mul_16_16(conj(cZ(3)),(cfB2))); cDENOM_32(1) = Cmplx_Add_32_32(Cmplx_Mul_16_16(conj(cZ(4)),(cfB1)),           Cmplx_Mul_16_16(conj(cZ(5)),(cfB2))); cMul_NUM_cDENOM_64(0) = Cmplx_Mul_32_32(cNUMERATOR_32(0),conj(cDENOM_32(0))); cMul_NUM_cDENOM_64(1) = Cmplx_Mul_32_32(cNUMERATOR_32(1),conj(cDENOM_32(1))); DENOM_2_64(0) = MAG_2_32(cDENOM_32(0)); DENOM_2_64(1) = MAG_2_32(cDENOM_32(1)); %Level Control for division scale denominator down to 32bit if(DENOM_2_64(0)>DENOM_2_64(1) )  Scale = ceil_log2(DENOM_2_64(0))); else  Scale = ceil_log2(DENOM_2_64(1))); end Scale = 31−Scale; If(Scale>=0) DENOM_2_32(0) = Cmplx_Saturate (DENOM_2_64(0)<<(Scale),31); DENOM_2_32(1) = Cmplx_Saturate (DENOM_2_64(1)<< (Scale),31); else DENOM_2_32(0) = Cmplx_Saturate( DENOM_2_64(0)>>(−Scale),31); DENOM_2_32(1) = Cmplx_Saturate (DENOM_2_64(1)>>(−Scale),31); end %complex/real division 64bit/32bit gives 32bit result cfCM(0) = Cmplx_real_div_64_32 (cMul_NUM_cDENOM_64(0)>>N2 ,DENOM_2_32(0)); cfCM(1) = Cmplx_real_div_64_32 (cMul_NUM_cDENOM_64(1)>>N2 ,DENOM_2_32(1)); cfCM_avg = Cmplx_Add_32_32 (cfCM(0),cfCM(1)); % Rotation of FCM _avg needed only if HW can not Insure Delatn=0 [cphasor_dn tmp] = cordic_SW( (CFO<<1)*N_delta, 0); cfCM_avg_rot_64 = Cmplx_Mul_32_16(cfCM_avg,cphasor_dn); cfCM_avg_rot_32 = fCM_avg_rot_64>>(Ntan); Scale_fCM = Ntan−1−N1+Scale_ratio−Scale−6−N2; fCM_one_level_32 = 1<<(Scale_fCM); cfp_m_32 = Cmplx_Real_Add_32_32( cfCM_avg_rot_32, −fCM_one_level_32); cfp_p_32 = Cmplx_Real_Add_32_32( cfCM_avg_rot_32, fCM_one_level_32); cNUMERATORf_64 = Cmplx_Mul_32_32(cfp_m_32,conj(cfp_p_32)); DENOMf_64 =   MAG_2_32 (cfp_p_32); %Level Control for division scale denominator down to 32bit signed Abs_real_Numerator64 = abs(real(cNUMERATORf_64); Abs_imag_Numerator64 = abs(imag(cNUMERATORf_64); if(Abs_real_Numerator64) > Abs_imag_Numerator64) Scale_fn =30 − ceil_log2(Abs_real_Numerator64); else Scale_fn =30 − ceil_log2(Abs_imag_Numerator64); end if(Scale_fn>=0) cNUMERATORf_32= cNUMERATORf_64<<(Scale_fn); else cNUMERATORf_32= cNUMERATORf_64>>(−Scale_fn); end %Level Control for division scale denominator down to 16bit unsigned Scale_fd =16 − ceil_log2(DENOMf_64); if(Scale_fd >=0) DENOMf_scaled_16= Cmplx_Saturate (DENOMf_64<<(Scale_fd),16); else DENOMf_scaled_16= Cmplx_Saturate (DENOMf_64>>(−Scale_fd),16); end % Division 32bit complex by 16bit real cg_exp_mTeta_M_16 = Cmplx_real_div_32_16 (cNUMERATORf_32,DENOMf_scaled_16); Scale_g = Scale_fd−Scale_fn; scale_inv = 14−Scale_fd+Scale_fn;

TABLE C-7 Phasor Estimation Variable Definition Name Size Comments I/O sumfB1SfB2S 16 bit cNUMERATOR_32 32 bit × 2 complex array cDENOM_32 32 bit × 2 complex array cMul_NUM_cDENOM_64 64 bit × 2 complex array DENOM_2_64 64 bit × 2 array DENOM_2_32 32 bit × 2 array cfCM 32 bit × 2 complex array cfCM_avg 32 bit complex cphasor_dn 32 bit complex cfs11 16 bit complex fs21 16 bit fs22 16 bit cfCM_avg_rot_64 64 bit complex cfCM_avg_rot_32 32 bit complex cfCM_one_level_32 32 bit cfp_m_32 32 bit complex cfp_p_32 32 bit complex cNUMERATORf_64 64 bit complex DENOMf_64 64 bit Abs_real_Numerator64 64 bit Abs_imag_Numerator64 64 bit cNUMERATORf_32 32 bit complex DENOMf_scaled_16 16 bit complex cg_exp_mTeta_M_16 16 bit complex O Scale_g 16 bit O scale_inv 16 bit O Scale_fd 16 bit Scale_fn 16 bit Scale_fCM 16 bit Scale 16 bit Function 12: Compensation_Params_Estimation

function [theta,rho, Scale_Q] = Compensation_Params_Estimation(cg_exp_mTeta_M_16,   scale_inv,Scale_g) Mue_bits=4; if(ProbeII_Num == 1) Mue =12; elseif(ProbeII_Num == 2) Mue = 8; else Mue = 4; End if(ProbeII_Num>=1)  % First Order Loop  Mue_1m = (16−Mue);  cg_ACC_32 = cg_exp_mteta <<(−Scale_g−Mue_bits);  cg_Delta_32 = (Cmplx_Mul_16_16 (cg_exp_mteta *cg_exp_mTeta_M_16))>>4;  cg_ACC_32 = Cmplx_Add_32_32 (Cmplx_Mul_32_16(cg_ACC_32,Mue_Fix_1m),           Cmplx_Mul_32_16(cg_Delta_32,Mue_Fix  ))>>(−Scale_g);  cg_exp_mteta = g_ACC_32;   % I/Q Correction Params Calculation  [theta,rho, Scale_Q] = Compute_Fix_Point_IQ_Coeffs(cg_exp_mteta,scale_inv,Scale_g,0); else   g_exp_mteta = cg_exp_mTeta_M_16;   %  I/Q Correction Params Calculation  [theta,rho, Scale_Q] = Compute_Fix_Point_IQ_Coeffs(g_exp_mTeta_M_16,scale_inv, Scale_g,1); end ProbeII_Num = ProbeII_Num+1;

TABLE C-8 Compensation_Params_Estimation variable definition Name Size Comments I/O ProbeII_Num 16 bit Reset Value is 0 NA Mue 16 bit Loop Gain NA Mue_1m 16 bit Complementary Loop NA Gain cg_ACC_32 32 bit complex cg_Delta_32 32 bit complex cg_exp_mteta 16 bit complex theta 16 bit Reset Value 2048 O rho 16 bit Reset Value 0 O Scale_Q  1 bit Reset Value 1 O Function 13: Compute_Fix_Point_IQ_Coeffs

function [theta,rho, Scale_Q] =  Compute_Fix_Point_IQ_Coeffs(cg_exp_mTeta_M_16,  scale_inv, scale_g,FirstTime) if(FirstTime)  scale_delta=0;  if (scale_inv>31)   scale_delta = scale_inv−31;   scale_inv=31;  end  scale_inv_log = scale_inv;  scale_delta_log = scale_delta;  scale_g_log = scale_g else  scale_inv = scale_inv_log;  scale_delta= scale_delta_log;  scale_g = scale_g_log; end real_g_exp_mTeta_M_16 = real(cg_exp_mTeta_M_16); imag_g_exp_mTeta_M_16 = imag(cg_exp_mTeta_M_16); if( real_g_exp_mTeta_M_16 > 1<< (−scale_g_log) )  Inv_real_g_exp_mTeta_M_16 = Cmplx_real_div_32_16  (1<<(scale_inv),real_g_exp_mTeta_M_16));  Scale_theta = −14+Teta_bits−1+scale_delta;  If(Scale_theta>=0)  theta = Inv_real_g_exp_mTeta_M_16<<(Scale_theta);  else  theta = Inv_real_g_exp_mTeta_M_16>>(−Scale_theta);  end  Rho_32 = Cmplx_Mul_16_16  (−imag_g_exp_mTeta_M_16,Inv_real_g_exp_mTeta_M_16);  Scale_rho = −Rho_bits+1+scale_inv;  If(Scale_rho>=0)  rho = Rho_32<<(Scale_rho);  else rho = Rho_32>>(−Scale_rho);  end  Scale_Q = 1; else  Scale_theta = scale_g_log +Teta_bits−1;  Scale_(—) rho = scale_g_log + Rho_bits −1;  If(Scale_theta>=0)  theta = real_g_exp_mTeta_M_16<<(Scale_theta);  else  theta = real_g_exp_mTeta_M_16>>(−Scale_theta);  end  If(Scale_rho>=0)  rho = (−imag_g_exp_mTeta_M_16)<<(Scale_rho);  else rho = (−imag_g_exp_mTeta_M_16)>>(−Scale_rho);  end   Scale_Q = 0; end

TABLE C-9 Compute_Fix_Point_IQ_Coeffs variable definition Name Size Comments I/O scale_inv_log 16 bit scale_delta_log 16 bit scale_inv 16 bit scale_delta 16 bit real_g_exp_mTeta_M_16 16 bit imag_g_exp_mTeta_M_16 16 bit Inv_real_g_exp_mTeta_M_16 16 bit Scale_theta 16 bit Scale_rho 16 bit Rho_32 32 bit theta 16 bit O rho 16 bit O Scale_Q  1 bit O Complex Math Operation Definitions

-   [C_(—)32_r,C_(—)32_i]=Cmplx_Add_(—)16_(—)16(A_(—)16_r,A_(—)16_i,B_(—)16_r,B_(—)16_i);     -   C_(—)32_r=A_(—)16_r+B_(—)16_r     -   C_(—)32_i=A_(—)16_i+B_(—)16_i -   [C_(—)32_r,C_(—)32_i]=Cmplx_Add_(—)32_(—)32(A_(—)32_r,A_(—)32_i,B_(—)32_r,B_(—)32_i);     -   C_(—)32_r=A_(—)32_r+B_(—)32_r     -   C_(—)32_i=A_(—)32_i+B_(—)32_i -   [R_(—)32]=Real_Add_(—)30_(—)30_T(A_(—)30,B_(—)30);     -   R_(—)32=A_(—)30+B_(—)30 -   [C_(—)64_r,C_(—)64_i]=Cmplx_Add_(—)64_(—)32(A_(—)64_r,A_(—)64_i,B_(—)32_r,B_(—)32_i);     -   C_(—)64_r=A_(—)64_r+B_(—)32_r     -   C_(—)64_i=A_(—)64_i+B_(—)32_i -   [C_(—)32_r,     C_(—)32_i]=Cmplx_Real_Add_(—)32_(—)16(A_(—)32_r,A_(—)32_i,B_(—)16);     -   C_(—)32_r=A_(—)32_r+B_(—)16     -   C_(—)32_i=A_(—)32_i -   [C_(—)32_r,     C_(—)32_i]=Cmplx_imag_Add_(—)32_(—)16(A_(—)32_r,A_(—)32_i,B_(—)16);     -   C_(—)32_r=A_(—)32_r     -   C_(—)32_i=A_(—)32_i+B_(—)16 -   [C_(—)32_r]=MAG_(—)2_(—)16(A_(—)16_r,A_(—)16_i)     -   C_(—)32_r=A_(—)16_r*A_(—)16_r+A_(—)16_i*A_(—)16_i -   [C_(—)64_r]=MAG_(—)2_(—)32(A_(—)32_r,A_(—)32_i)     -   C_(—)64_r=A_(—)32_r*A_(—)32_r+A_(—)32_i*A_(—)32_i -   [C_(—)32_r,C_(—)32_i]=Cmplx_Mult_(—)16_(—)16(A_(—)16_r,A_(—)16_i,B_(—)16_r,B_(—)16_i)     -   C_(—)32_r=A_(—)16_r*B_(—)16_r_A_(—)16_i*B_(—)16_i     -   C_(—)32_i=A_(—)16_r*B_(—)16_i+A_(—)16_i*B_(—)16_r -   [C_(—)64_r,C_(—)64_i]=Cmplx_Mult_(—)32_(—)32(A_(—)32_r,A_(—)32_i,B_(—)32_r,B_(—)32_i)     -   C_(—)64_r=A_(—)32_r*B_(—)32_r_A_(—)32_i*B_(—)32_i     -   C_(—)64_i=A_(—)32_r*B_(—)32_i+A_(—)32_i*B_(—)32_r -   [C_(—)32_r,C_(—)32_i]=Cmplx_real_Mult_(—)16_(—)16(A_(—)16_r,A_(—)16_i,B_(—)16_r)     -   C_(—)32_r=A_(—)16_r*B_(—)16_r     -   C_(—)32_i=A_(—)16_i*B_(—)16_r -   [C_(—)64_r,C_(—)64_i]=Cmplx_Mul_(—)32_(—)16(A_(—)32_r,A_(—)32_i,B_(—)16_r)     -   C_(—)64_r=A_(—)32_r*B_(—)16_r     -   C_(—)64_i=A_(—)32_i*B_(—)16_r -   [C_(—)16_r,C_(—)16_i]=Cmplx_real_div_(—)32_(—)16(A_(—)32_r,A_(—)32_i,B_(—)16_r)     -   C_(—)32_r=A_(—)32_r/B_(—)16_r     -   C_(—)32_i=A_(—)32_i/B_(—)16_r -   [C_(—)32_r,C_(—)32_i]=Cmplx_real_div_(—)64_(—)32(A_(—)64_r,A_(—)64_i,B_(—)32_r)     -   C_(—)32_r=A_(—)64_r/B_(—)32_r     -   C_(—)32_i=A_(—)64_i/B_(—)32_r -   [C_(—)16_r,C_(—)16_i]=Switch real_imag(A_(—)16_r,A_(—)16_i);     -   C_(—)32_r=A_(—)16_i     -   C_(—)32_i=A_(—)16_r

APPENDIX D

Exemplary Parameters of a HW-SW Interface

The following table summarizes the information exchanged between the HW and SW during probe2 reception. Output refers to Output from the HW and Input refers to Input to the HW.

Input to/Output from HW Name SW Name Size HW Remarks Δn N_delta 16 bits Output Number of samples between NCO reset and first sample of the FFT window. i1 I1  3 bit Output Offset index of first negative FFT bin from nominal i2 I2  4 bit Output Offset index of second negative FFT bin from nominal CFO CFO 17 bit Output Carrier Frequency Offset (after fine frequency offset) Z[k1,:] cIQparameters_log(0,:) 16 × 2 × Nprobe2_symbols bit Output FFT output at bin k1 for symbols 1, . . . , Nprobe2_symbols Z[k2,:] cIQparameters_log(1,:) 16 × 2 × Nprobe2_symbols bit Output FFT output at bin k2 for symbols 1, . . . , Nprobe2_symbols Z[−k1 + i1,:] cIQparameters_log(2,:) 16 × 2 × Nprobe2_symbols bit Output FFT output at bin −k1 + i1 for symbols 1, . . . , Nprobe2_symbols Z[−k1 + i2,:] cIQparameters_log(3,:) 16 × 2 × Nprobe2_symbols bit Output FFT output at bin −k1 + i2 for symbols 1, . . . , Nprobe2_symbols Z[−k2 + i1,:] cIQparameters_log(4,:) 16 × 2 × Nprobe2_symbols bit Output FFT output at bin −k2 + i1 for symbols 1, . . . , Nprobe2_symbols Z[−k2 + i2,:] cIQparameters_log(5,:) 16 ×2 × Nprobe2_symbols bit Output FFT output at bin −k2 + i2 for symbols 1, . . . , Nprobe2_symbols ζ theta 12 bit unsigned Input I/Q Compensation parameter ρ Rho 12 bit Input I/Q Compensation parameter Scale_Q Scale_Q  1 bit Input I/Q Compensation parameter

APPENDIX E

Frequency Offset Introduction, CP Length Number of Symbols

function [dF] = Frequency_Offset_Introduction(CFO) Freq_Th = 41;        %round((5e3/50e6)*2*pi*2{circumflex over ( )}16) if (abs(CFO)< Freq_Th)  dF = sign(CFO)*(TBD RF Interface introduce); else   dF=0; end Tone Selection

function [SC] = ProbeII_Tone_Selection(Sigma_2_32) SC_MIN(0) = 146; SC_MAX(0) = 186; SC_MIN(2) = 217; SC_MAX(2) = 249; SC_DEFAULT(0) = 176; SC_DEFAULT(1) = 249; NLog_In_Bits=6; NLog_Out_Bits=14; SC = SC_DEFAULT; a=−7871; % round(−1.921625277102556*2{circumflex over ( )}(NLog_Out_Bits−2) ) ; b= 8071; % round( 1.970377382221271*2{circumflex over ( )}(NLog_Out_Bits−2) ); C =32656; %round(0.0375*log2(10)*2{circumflex over ( )}(NLog_Out_Bits−2 + NLog_In_Bits)); %Find Default NSR log2(Sigma_2) for i=0:1  Scale_16(i) = ceil_log2( Sigma_2_32(SC_DEFAULT(i)) );  Frac_16(i) = Sigma_2_32(SC_DEFAULT(i))>> (Scale_16(i) −NLog_In_Bits );  NSR_Default_32(i) = Scale_16(i) << (NLog_Out_Bits − 2 + NLog_In_Bits)+ ...       (Real_Mult_16_16(b,Frac_16(i))+a<<(NLog_In_Bits) );  NSR_Best_32(i) = NSR_Default_32(i);  for k=[SC_MIN(i) : SC_MAX(i)]   Scale_k_16 = ceil_log2( Sigma_2_32 (k) );   Frac_k_16 = Sigma_2_32 (k)>>(Scale_k_16 −NLog_In_Bits );   NSR_32 = Scale_k_16 << (NLog_Out_Bits−2 + NLog_In_Bits)+ ...       (Real_Mult_16_16(b,Frac_k_16)+a<<(NLog_In_Bits) );   if(NSR_32< Add_Real_32_32(NSR_Default_32 (i),...       − Real_Mult_16_16(C, Abs_16(Add_Real_16_16(k, −SC_DEFAULT(i)) )    if(NSR_32 < NSR_Best_32(i))     SC(i) = k;     NSR_Best_32(i) = NSR_32;    end   end  end end Where Sigma_32 is a vector of the estimated noise variance of the various FFT tones. CP & Number of OFDM Symbol Selection

function [NUM_OF_SYMS, CP_LENGTH] = Probe2_CP_L_Select(CFO) CP_Max = 126; CP_Min = 64; L_Max = 40; L_Min = 28; PI = 205887; %round(pi*2{circumflex over ( )}Freq_bits−1) 2P = 411775; %round(2pi*2{circumflex over ( )}Freq_bits−1) Phase_Th = 41177; %round(2pi/10*2{circumflex over ( )}Freq_bits−1) dPhase_64= 0; error= 0; Min_Err = Inf; NUM_OF_SYMS = L_Min; CP_LENGTH =CP_Min; for CP = CP_Min: CP_Max  for L= L_Min: L_Max   dPhase_32 = L*(CP+Nfft)*(CFO);   while (abs(dPhase_32) > 2P)    dPhase_32= dPhase_32− sign(dPhase_32)* 2P;   end   if (abs(dPhase_32 )>PI)    error_32 = abs(2PI − abs(dPhase_32));   else    error_32 = abs(dPhase_32);   end   if( error_32 < Phase_Th )    Indicator=1;    if(error_32 <Min_Err)     Min_Err = error;     NUM_OF_SYMS = L;     CP_LENGTH =CP;    end   end  end end 

1. A system for compensating for an imbalance between a first signal and a second signal, the system comprising: a first module operative to record, based on the first and second signals, a first frequency domain parameter and a second frequency domain parameter corresponding to each of a first tone, a second tone and a carrier frequency; and a second module operative to compute at least one time domain compensation parameter based on the first and second frequency domain parameters, wherein: the carrier frequency is a receiver carrier frequency; the first and second signals are received from a transmitter operative to transmit the signals using a transmitter carrier frequency; the receiver and transmitter carrier frequencies differ by a carrier frequency offset the first and the second frequency domain parameters correspond to one of the first and the second tones: the first frequency domain parameter corresponds to a first bin in a discrete-valued frequency spectrum; the second frequency domain parameter corresponds to a second bin in the spectrum; the first bin is contiguous with the second bin; the first module is operative to select the second bin based on the location in the spectrum of the first bin and based on the ratio of the offset, in radians, to Pi radians.
 2. The system of claim 1 further comprising a third module operative to output a compensated signal based on the time domain compensation parameter.
 3. The system of claim 1 wherein the at least one time domain compensation parameter comprises three time domain compensation parameters.
 4. The system of claim 1 wherein, when there is a residual carrier frequency offset, the second module is operative to: compute an estimate of the residual frequency offset; and compute the time domain compensation parameter based on the estimate.
 5. The system of claim 1 wherein, when the first module is operative to record the first and second frequency domain parameters for each of a plurality of symbols that are received serially in time, the second module is operative to compute a first average value for the first frequency domain parameter and a second average value for the second frequency domain parameter, the average values based, respectively, on the first and second frequency domain parameters corresponding to the symbols.
 6. The system of claim 1 wherein: the first module is a hardware module; and the second module is a software module.
 7. The system of claim 2 wherein: the first module is a hardware module; the second module is a software module; and the third module is a hardware module.
 8. The system of claim 1 wherein: the first module is a hardware module; and the second module is a hardware module.
 9. The system of claim 2 wherein the third module is a hardware module. 